Velocity field, surface profile and curvature resolution of steep and short free-surface waves

Abstract

On steep, millimeter-scale, 2D water waves, surface profile, and subsurface velocity field are measured with high-spatio-temporal resolution. This allows resolving surface vorticity, which is captured in the surface boundary layer and compared with its direct computation from interface curvature and velocity. Data are obtained with a combination of high-magnification time-resolved particle image velocimetry (PIV) and planar laser-induced fluorescence. The latter is used to resolve the surface profile and serves as a processing mask for the former. PIV processing schemes are compared to optimize accuracy locally, and profilometry data are treated to obtain surface curvature. This diagnostic enables new insights into free-surface dynamic, in particular, wave growth and surface vorticity generation, for flow regimes not studied previously. The technique is demonstrated on a high-speed water jet discharging in quiescent air at a Reynolds number of 4.8 × 10⁴. Shear-layer instability below the surface leads to streamwise traveling waves with wavelength λ ~ 2 mm and steepness $2\pi a / \lambda \ge 2.0$, where a is the crest to trough amplitude. Flow structures are resolved at these scales by recording at 16 kHz with a magnification of 4.

Appendices

Appendix 1

In order to find the relation between γ ₁, γ ₂, and g/w, a spherical coordinate system is used to represent all the possible spanwise and streamwise angles. $\phi$ and $\psi$ are respectively the polar and azimuthal angles of the spherical coordinate system shown in Fig. 23.

Fig. 23

Spherical coordinate system. v is the viewing vector pointing toward the PLIF camera

v is the unit vector pointing toward the PLIF camera and n is the local surface normal unit vector. γ ₁ is defined as the angle between the axis z and the projection of n in the plane containing x and z, and its expression is given by Eq. 19:

$$\begin{aligned} \gamma _1 = \arctan \left( \sin (\phi )\tan (\psi )\right) \end{aligned}$$

(19)

γ ₂ is defined as the angle between the axis y and n, minus $\pi /2$, and its expression is given by Eq. 20:

$$\begin{aligned} \gamma _2 = \arcsin \left( \sin (\psi )\cos (\phi )\right) \end{aligned}$$

(20)

The apparent thickness of the gradient zone, as seen by the camera is given by Eq. 21:

$$\begin{aligned} g= \frac{w}{\cos (\gamma _2)} \mathbf{v} \cdot \mathbf{n} \end{aligned}$$

(21)

which can be rewritten

$$\begin{aligned} \frac{g}{w} =\frac{\cos (\alpha _2) \cos (\phi ) \sin (\psi )+ \sin (\alpha _2) \cos (\psi )}{\cos (\gamma _2)} \end{aligned}$$

(22)

By plugging Eq. 20 into 22, the system reduces to a second degree polynomial function whose variable is $\cos (\phi )$ and whose coefficients depend on g/w and $\psi$. Once $\psi$ and $\phi$ are known for a given g/w, γ ₁ and γ ₂ can be calculated a posteriori. Direct calculation of γ ₂ from γ ₁ and g/w can be done graphically using Fig. 8 or requires an iterative process.

Appendix 2

The surface profile can be approximated in the spanwise direction by a second order Taylor expansion:

$$\begin{aligned} f\left(x,y+\frac{w}{2}\right)&=f(x,y) + \frac{w}{2} \tan (\gamma _2) \nonumber \\&\quad+ \frac{w^2}{8} \kappa _y \left( 1 + (\tan (\gamma _2))^2\right) ^{3/2}+O(w^3) \end{aligned}$$

(23)

The change in elevation due to the spanwise curvature, $\kappa _y$, can be estimated using the second order term of Eq. 23. The error due to this term, $\epsilon _y$ is plotted in Fig. 24. Additionally, curvature can limit the angle at which the spanwise profile can be entirely seen. The limit is given by Eq. 24, and the region above this limit is shown in gray in Fig. 24.

$$\begin{aligned} |\kappa _y| =\frac{4 (\tan (\alpha _2)+\tan (\gamma _2))}{w \left( 1 + (\tan (\gamma _2))^2\right) ^{3/2}} \end{aligned}$$

(24)

Fig. 24

Iso-contour of $\epsilon _y/w$ (in %) showing the error in the surface elevation induced by neglecting the second order term (spanwise surface curvature). Region where the interface is masked due to the curvature is in gray

Appendix 3

In the present coordinate system for a 2D flow, the flux of surface parallel vorticity through the surface is given by Eq. 25 (Rood 1995; Gharib and Weigand 1996). Note that the out of plane vector s × n is $-\mathbf{y}$ by consistency with the coordinate system defined in Figs. 1 and 2.

$$\begin{aligned} \nu \mathbf{n} \cdot \nabla \omega = \left( \frac{\partial u_s}{\partial t}-u_n \omega _{n=0} +\frac{1}{2}\frac{\partial (u_s^2 + u_n^2)}{\partial s}+\frac{1}{\rho }\frac{\partial p}{\partial s}-g \sin \gamma _1 \right) \mathbf{y} \end{aligned}$$

(25)

The pressure gradient term can be obtained based on the surface normal stress boundary condition (Lundgren and Koumoutsakos 1999):

$$\begin{aligned} p =p_{\rm gas}+ \sigma \kappa -2 \mu \left( \frac{\partial u_s }{\partial s} + u_n \kappa \right) , \end{aligned}$$

(26)

where $p_{\rm gas}$ is the pressure on the gas side of the interface, which can be assumed constant.

The five terms contributing to the flux are surface acceleration, advected surface vorticity, advective acceleration, pressure gradient and gravity, in the same order as they appear in the right hand side of Eq. 25. According to Dabiri and Gharib (1997), the net vorticity flux through the surface $\zeta$, can be calculated with the running integral of Eq. 25 with respect to s as following:

$$\begin{aligned} \zeta = -\int\limits _{s_0}^{s} \nu (\mathbf{n} \cdot \nabla \omega )\cdot \mathbf{y} {\rm d}s \end{aligned}$$

(27)

It should be noted that a flux of positive vorticity into the liquid and a flux of negative vorticity out of the liquid will both result in $-(\nu \mathbf{n} \cdot \nabla \omega ) \cdot \mathbf{y} > 0$. The temporal evolution of the bulk vorticity can be used to solve this ambiguity.