Effects of air-side freestream turbulence on the development of air–liquid surface waves

Abstract

It is important to investigate the effects of air-side freestream turbulence on the development of air–liquid surface waves. Two turbulent-grid air flows are generated by regular and fractal grids in a wind-wave tank. Wind velocity and water-level fluctuations are measured with and without grids. The results show that regular and fractal grids can generate turbulent-grid air flows with different wind velocity fluctuations. The spectra of water-level fluctuations in the regular-type and fractal-type grid cases are consistent with those of pure wind-driven waves without a grid. Furthermore, as wind-wave properties, including the fetch law, dispersion relation, and Toba’s 3/2 power law, all hold in cases with and without grids, it is confirmed that the waves develop through the turbulence only near the free water surface, and not through the air-side freestream turbulence. This suggests that the effects of air-side freestream turbulence on wave development are negligible.

Graphic abstract

Appendices

Appendix 1

The vertical velocity, $\bar{W}$, averaged over the whole span of the tank should be zero by continuity. However, it is − 0.11 m s⁻¹ with and without grids at z = 75 mm (Fig. 5) when averaged over the more limited measurement range (− 100 < y < 100 mm). This may be a result of using an incomplete span or, more likely, a result of measurement error due to the camera tilt. The deduced tilt is small (θ = 0.95°). The following describes the measurement error caused by the camera tilt.

The tilt θ is estimated by the following equation:

$$ \theta = \tan^{ - 1} \left( { - \frac{{\bar{W}}}{{\bar{U}}}} \right), $$

(7)

where $\bar{U}$ shows the mean streamwise velocity at z = 75 mm. The $\bar{U}$ in Fig. 4c is approximately 12 m s⁻¹, and $\bar{W}$ is approximately − 0.11 m s⁻¹ in Fig. 5, with and without grids. Here, Eq. (8) is provided from the vector rotation:

$$ \left( {\begin{array}{*{20}c} {\overline{{U^{\prime } }} } \\ {\overline{{W^{\prime } }} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\bar{U}} \\ {\bar{W}} \\ \end{array} } \right), $$

(8)

where $\overline{{U^{\prime } }}$ and $\overline{{W^{\prime } }}$ show rotated $\bar{U}$ and $\bar{W}$, respectively. It is difficult to avoid camera tilt entirely in PIV setups. If we rotate the vertical velocity using the vector-rotation procedure with θ = 0.95°, we can provide the distributions of the rotated vertical velocity. The rotated vertical velocities are demonstrated with θ = 0.95° in Fig. 15. The secondary flows shown in Fig. 15 are similar as in Fig. 5 both with and without grids, but $\bar{W}$ is zero in Fig. 15.

Fig. 15

Vertical and spanwise distributions of rotated mean vertical velocity with and without grids at − 100 < y < 100 mm and 50 < z < 100 mm and x = 4.4 m. Vertical velocities are rotated using the vector-rotation procedure (Eq. 8). Top, middle, and bottom figures show the vertical velocity at NG, RG, and FG cases, respectively. Wind speed is 14.0 m s⁻¹. Data are measured by PIV

Appendix 2

In this experiment with the wind-wave tank, the mean water level at the measurement location (x = 4.4 m) changes due to static pressure (e.g., Liberzon and Shemer 2011; Caulliez et al. 2008). We measured the mean water levels with and without airflow using a water-level gauge (see Fig. 16). The water surface in the gauge was indicated by a floating red ball. The motion of the ball and the water surface in the gauge was captured by a CMOS camera (Nikon, D5100) with a lens (Nikon, AI AF Nikkor 20 mm f/2.8D). The sampling time was 180 s, and the sampling frequency was 1 Hz. The spatial resolution of each image was 0.085 mm pixel⁻¹. The motion of the ball was automatically tracked using motion-tracking software (Photron FASTCAM Analysis), and the mean water level (h_w) in the gauge was determined. The mean water level (h) in the wind-wave tank was estimated by h = h_w + h_b − h_a, where h_a (= 0.7 m) was the distance between the wind-wave tank and the ground, and h_b (= 1.231 m) was the distance between the water-level gauge and the ground. Verification experiments were performed for NG cases.

Fig. 16

Water-level gauge. h₀: stilled water level in the wind-wave tank; h: mean water level affected by wind pressure at the measurement point (x = 4.4 m); h_a: distance between the wind-wave tank and the ground; h_b: distance between the water level the gauge and ground; h_w: mean water level in the water-level gauge

The results showed that h increased due to static pressure, and the differences between h and the initial water level (h₀) without static pressure were 0.1, 0.6, and 1.8 mm at U_∞ = 6.4, 10.1, and 14.0 m s⁻¹, respectively. To estimate the friction velocities ($u_{2}^{*}$), considering the change of mean water level due to air-side pressure, the vertical coordinate (z₂) was corrected using h, i.e., z₂ = z − (h − h₀). Then, $u_{2}^{*}$ was estimated by $u_{2}^{*} = \lim_{{z_{2} \to 0}} \sqrt { - \overline{{u^{\prime } w^{\prime } }} }$. The results were found to be that $u_{2}^{*}$ was 0.1, 0.6, and 1.5% higher than $u^{*}$ at U_∞ = 6.4, 10.1, and 14.0 m s⁻¹, respectively. Here, $u^{*}$ was the uncorrected friction velocity value. Thus, we can conclude that the change in the mean water level at the measurement location is negligible in the experiments.