Bichromatic synthetic schlieren applied to surface wave measurements

Abstract

The present paper provides an introduction to bichromatic synthetic schlieren (BiCSS) for surface measurements, a novel extension of the free surface synthetic schlieren (FS-SS) by Moisy et al. (Exp Fluids 46:1021–1036, 2009). The new technique is based on the fact that light diffraction through a medium varies with wavelength. Therefore, one may apply light at two different wavelengths to measure the change in density gradient in a medium. This paper explores the use of the difference between blue visual and near-infrared light, but the choice of wavelengths will typically depend on the application. Calibration was performed using stationary targets of plexiglass and the results show that the new BiCSS technique improves accuracy for large surface gradients, compared to the traditional FS-SS technique. In order to test the applicability of the technique in the laboratory, two sets of experiments were performed. Firstly, an experiment using phase-locked regular waves was conducted for comparing BiCSS with FS-SS, analyze the properties and give an estimation of the error. Secondly, to investigate the applicability for more complex surface patterns, a study on a vertical surface-piercing cylinder exposed to a focused wave was conducted, obtaining the complex surface characteristics. The new technique clearly reveals nonlinear wave diffraction, in addition to cross waves and parasitic capillary waves.

Appendix A: Wave damper

Due to the directional wave propagation assumptions in determination of the integration constant, it is of importance to reduce reflection from the end wall of the tank. Beaches, while excellent for deep water waves, do not perform equally well when going towards shallow water. Instead a 3D printed wave damper consisting of many of narrow $L =15$ cm tall spikes with bases arranged using Penrose tiling was used. To evaluate the efficiency of the wave damper, the reflection index was estimated using a method by Goda and Suzuki (1977)

$$\begin{aligned}&RI = \left| \frac{\mathcal {A}}{\mathcal {B}}\right| ,\quad \text { where } {\mathcal {A}} = \frac{\hat{\eta }(x,\omega ) e^{ik_0\Delta x} - \hat{\eta }(x+\Delta x,\omega )}{e^{ik_0\Delta x}-e^{-ik_0\Delta x}} \text { and } \nonumber \\&\quad {\mathcal {B}} = \frac{\hat{\eta }(x+\Delta x,\omega ) - \hat{\eta }(x,\omega )e^{-ik_0\Delta x}}{e^{ik_0\Delta x}-e^{-ik_0\Delta x}}. \end{aligned}$$

(21)

The range of the reflection index was extended with respect to singularities by using five ultrasonic wave probes placed at $x =$ 0, 8, 12, 18 and 24 cm. This creates a set with distances $\Delta x =$ 4, 6, 8, 10, 12, 16, 18 and 24 cm between pairs, and the refraction index is estimated as a weighted combination of all possible pair of probes, shown in Fig. 11.

Fig. 11

Reflection index for spiked wave damper. $f_{min}$ and $f_{max}$ indicates the singularity of the estimation method

The wave damper has minimal reflection at $\lambda \approx L$, but it should be noted that poor signal-to-noise ratio at the higher frequency range makes the estimate less accurate. In addition the estimate will also be influenced by the singularities at the end points of the range. The regular waves used in the phased-locked error estimation has a frequency $f/\sqrt{g/L}=0.3710$, at this frequency the wave damper works reasonable well with $RI\approx 0.1$. For the focus waves the center frequency $f_0/\sqrt{g/L}=0.2879$, in this case the reflection is of less importance since the measurement is taken before the reflection reaches the measurement section.