Scanning stereo-PLIF method for free surface measurements in large 3D domains

Figure 2 shows the experimental set-up used in this study. The experiments are performed in the water tunnel of the Laboratory for Aero- and Hydrodynamics at the Delft University of Technology. The test section has an area of \(0.6\times 0.6\) m\(^2\) and a length of 5 m. A false bottom, 190 mm above the channel bottom, is used to generate a defined boundary layer, and to allow the water tunnel to be operated at a reduced water depth.

Free surface waves are generated behind an obstacle, which is mounted on the false bottom at a distance of 0.85 m from the leading edge. The shape of the obstacle is defined by a fourth-order polynomialwith \(H_b = 0.117\) m the obstacle height, and \(L = 0.295\) m the obstacle half-length (Gui et al. 2014).

In the current work, a number of free surface wave patterns is obtained by varying the Froude number, while keeping the initial water depth constant. The free surface wave pattern is defined by the water depth above the obstacle (\(H_0-H_b\)) and the upstream Froude number \(\text {Fr}=U_0/\sqrt{gH_0}\), where \(U_0\) is the upstream bulk velocity, g is the gravitational acceleration, and \(H_0\) is the initial water depth. The upstream Froude number is always sub-critical (\(\text {Fr}<1\)). However, transition to critical conditions (\(\text {Fr}=1\)) occurs at or near the obstacle (Gui et al. 2014).

The upstream flow conditions, at constant water depth (\(H_0\)), are determined prior to the free surface wave height measurements (Table 2). The liquid velocity is measured with a disk-type programmable electromagnetic liquid velocity meter (P-EMS E30, Deltares) where one of the axes is aligned with the flow. The probe is calibrated for a velocity range of \(\pm \, 1\,{\mathrm {m~s}}^{-1}\) with an accuracy of 1\(\%\). The initial water depth (\(H_0\)) is measured with a ruler.

A commonly applied PLIF system is used as a reference for the stereo-PLIF free surface measurements (Buckley and Veron 2016; Duncan et al. 1999). The light sheet from a Nd:YLF laser (LDY 304PIV laser, Litron) illuminates the liquid containing a fluorescent dye (Rhodamine WT at \(120 \, {\mathrm {mg~m}}^{-3}\)). The concentration of fluorescent dye is low enough such that it does not influence the static surface tension of the fluid (Appendix 1). The dynamic surface tension is more appropriate for steep waves where compression can locally alter the surfactant concentration (Duncan et al. 1999). However, for the current application, the static surface tension is sufficient, but the effect of compression on the surfactant concentration at a time scale similar to the wave action needs to be investigated in future work. Images are acquired with a high-speed CMOS camera (Imager HS 4M, LaVision) equipped with a \(105\,\mathrm {mm}\) Micro-Nikkor objective and a high-pass filter (\(\mathrm {OG}570\), Schott). The magnification (\(M_0\)) at the center plane is about \(M_0 = 0.1\), with an object distance of \(Z = 1\, \mathrm {m}\). A large depth-of-field is obtained, which with an aperture of f / 11, and a wavelength of \(\lambda = 527\, \mathrm {nm}\), results in \(\delta z \cong 4(1+1/M_0)^2 f^{\#^2} \lambda \approx 30\, \mathrm {mm}\) (Adrian and Westerweel 2011). The camera (Fig. 2) is placed at an angle (\(\beta \)) of \(15^{\circ }\) with respect to the light sheet (xy-plane) to avoid interference from the liquid meniscus (Belden and Techet 2011).

An inverse, third-order polynomial is used to determine the mapping from image to world coordinates (Soloff et al. 1997). The camera is calibrated using a two-plane dot-pattern target (Type 22, LaVision). A resolution of 10.1 pixels per millimeter is obtained over a field-of-view of approximately \(180\times 180 \, {\mathrm {mm}}^2\).

The stereo-PLIF system extends a PLIF system to a three-dimensional domain with a scanning light sheet and a stereo-camera set-up. The method is comparable to conventional techniques such as scanning-PIV (Brücker 1996). Images are acquired with two high-speed CMOS cameras (Imager HS 4M, LaVision) equipped with a \(105\,\mathrm {mm}\) Micro-Nikkor objective and a high-pass filter (\(\mathrm {OG}570\), Schott). The magnification at the center plane is approximately \(M_0 = 0.1\), which with an aperture of f/11 results in a depth-of-field of \(\delta z \approx 30 \, \mathrm {mm}\).

The cameras are placed in a stereo configuration (Fig. 2), with a full separation angle (\(2\alpha \)) of either \(30^{\circ }\) or \(50^{\circ }\). These two full separation angles are imposed by optical limitations in the experimental facilities, respectively, the Multiphase Wave Lab¹ at MARIN in The Netherlands and the water tunnel of the Laboratory for Aero- and Hydrodynamics at the Delft University of Technology. Optimal accuracy, for conventional stereo-PIV applications, is obtained for a full separation angle of \(60^{\circ } \le 2\alpha \le 90^{\circ } \) (Lawson and Wu 1997). The two angles are used to determine the influence of the separation angle on the measurement accuracy at comparable focal points (\(L_f \approx 1 \, \mathrm {m}\)). Consequently, the nominal magnification remains the same for both camera separation angles. Therefore, the error is expected to scale as \(e_r \propto \sigma _{{\varDelta }z}/\sigma _{{\varDelta }x} \propto \tan (\alpha )^{-1}\) (Lawson and Wu 1997).

Two-dimensional free surface height measurements are obtained by scanning the light sheet in a sawtooth profile over the three-dimensional measurement domain with an oscillating mirror driven by a galvanometer (\(\mathrm {CT-}6210\mathrm {H}\), Cambridge Technology) located at \(y_c = 1745 \, \mathrm {mm}\) above the false bottom (Fig. 2a). The large distance of the scanning mirror ensures almost vertical measurement planes (Fig. 5a), with a maximum angle deviation, with respect to the y-axis, of approximately \(2.3^{\circ }\) at the outer edges of the measurement domain (\(z_d \approx 62 \ \mathrm {mm}\)) (Fig. 5b).

An accurate edge detection method is required for the calibration procedure and the free surface reconstruction. Large variations in image intensity (I) are observed as a result of the liquid properties (Fig. 3a). The variations are a result of refraction, light focusing, and reflection at the air–water interface. Traditional edge detection can result in detection errors due to the semi-reflective properties of the air–water interface (André and Bardet 2014).

The edge detection error depends also on the camera separation angle. The specular bias is reduced at small camera separation angles, but the quantization errors increase. Hence, there is trade-off between detection errors and quantization errors (Benetazzo 2006). Furthermore, the variation in intensity along the laser light sheet can introduce other angle dependent errors (André and Bardet 2014). These errors can be reduced with other detection methods, such as hyperbolic tangent fit methods (Hwung et al. 2009). However, the detection accuracy is not necessarily improved, and often these methods require an increased computational effort. Instead, a multi-step intensity-based detection procedure is used in the current work.

The edge information is obtained from the raw image (\(2016\times 2016 \, \mathrm {pixels}\)) with a multi-step intensity-based detection procedure (Fig. 3a). First, the background intensity variation, based on the windowed mean and standard deviation, is determined over the first part of the image (\(500 \, \mathrm {pixels}\)). Then, a discontinuous threshold distribution is determined per window, which is defined as the mean plus two times the standard deviation (Fig. 3b). Finally, a continuous threshold distribution is obtained by fitting a second-order polynomial to the discontinuous data.

The raw image is binarized with the continuous threshold distribution. Then, the image is smoothed with a two-dimensional \(3\times 3\) Gaussian filter to suppress the small-scale image noise (Fig. 3c). Finally, the edge is defined as the pixel with a gray level above the threshold with the lowest y-value (i.e. topmost “white” pixel in Fig. 3c).

Detection errors can occur with a per-column threshold operation. However, pixel data cannot be easily filtered, due to the variation in magnification over the domain. This would result in a variable filter length in world coordinates. Therefore, pixel coordinate data are processed without further filtering.

The scanning stereo-PLIF system is calibrated with a self-calibration procedure after an initial calibration at the outer edges of the measurement domain (i.e., front and aft plane). An updated mapping function is determined at each light sheet location for the two-dimensional free surface measurements.

The self-calibration procedure is similar to typical stereo-PIV applications (e.g., Hori and Sakakibara 2004; Wieneke 2008). There is, however, a distinct difference. A stereo-PLIF measurement contains only curvilinear lines (i.e., the intersection of the light sheet with the surface) at a specific free surface height, whereas a stereo-PIV measurement contains information over the entire field-of-view. Images of the liquid free surface are obtained at several unique still water heights, hereby sampling the entire field-of-view. The whole process of the stereo-PLIF calibration procedure is detailed in the following sections.

First, a broad calibration is performed at two planes enclosing the measurement domain (i.e., positive and negative z), shown in Fig. 4, with a two-plane dot pattern target (Type 22, LaVision). An inverse third-order polynomial mapping function \(\mathbf {x} = F_{0_{i,j}}^{-1}(\mathbf {X})\) is determined for each camera (j) and plane (i), which maps pixel coordinates \(\mathbf {X} = (X,Y)\) to world coordinates \(\mathbf {x} = (x,y,z)\) (Soloff et al. 1997).

Then, images of the free surface are acquired at several still water heights (\(N=14\)) for each light sheet location (k). The edge detection procedure is applied to obtain curvilinear lines (\(L_{k,n}\)) for each light sheet location (k) and still water height (n). These lines span the entire field-of-view.

Compared to stereo-PIV, the lines do not contain identifiable points (i.e., matching world and pixel coordinates). Furthermore, identifiable points can not be obtained with the approach of Hori and Sakakibara (2004), as the scanning mirror is not calibrated. Identifiable points are obtained by distance minimization between camera projection lines (\(R_j(\mathbf {x})\)), which are the rays formed by back-projection of single pixels to the enclosing calibration planes (Fig. 4). The projection lines of both cameras are matched based on the minimal distance between skew lines with a maximum distance threshold (Gellert et al. 1989). The point of minimal distance defines the world coordinate (\(\mathbf {x}\)) with corresponding pixel coordinates for both cameras (Fig. 4).

Next, an updated inverse polynomial mapping is determined for each camera and light sheet location (k). The matched features are all constrained on a light sheet (i.e., a single plane), which enables the use of back-projection methods. The inverse mapping defines world coordinates \((x,y,0) = F^{-1}_{j,k}(\mathbf {X})\) on a light sheet plane (Adrian and Westerweel 2011). The mapping function is completed by fitting a plane to the z-coordinates (\(z = f(x,y)\)).

Finally, the calibration domain is aligned with the liquid free surface, as the rough calibration is not necessarily aligned with the still water level. The two-dimensional free surface height \(y = y(x,z)\) is determined with the calibrated stereo-PLIF system. A plane is fitted to the reconstructed two-dimensional free surface, which is subtracted as a correction for the misalignment.

Post-processing is applied to the world coordinates obtained with both stereo cameras. The post-processing is applied to obtain uniformly distributed coordinates. Furthermore, the obtained data are filtered and averaged to obtain a combined stereo-PLIF measurement.

First, the pixel data obtained with the edge processing procedure are mapped to the respective measurement plane (i.e., light sheet location k) using the updated mapping function. The stereo camera data cannot be directly combined as the world points are non-uniformly distributed. Therefore, the world data are interpolated, with a linear interpolation method, to a uniformly distributed grid over the x-direction with a spacing of \({\varDelta }x = 0.1 \, \mathrm {mm}\). Higher-order interpolation methods are in the current application not required, as a grid point is displaced by only \(||\mathbf {x} - \mathbf {x}_i|| \approx 0.04 \, \mathrm {mm}\) on average.

Then, the now uniformly distributed stereo data is filtered. First, a Hampel filter, with a filter length of \(L_{h,f} = 2.7 \, \mathrm {mm}\), is applied, which removes values that deviate more than three standard deviations from the median over the filter length (Liu et al. 2004). Then, to smooth the data, a second-order Savitzky–Golay finite impulse response filter is applied, with a filter length of \(L_{h,s} = 2.7 \, \mathrm {mm}\) (Orfanidis 1995). Finally, the stereo-PLIF data are obtained by averaging the data from both stereo cameras.

The resolution of the initial polynomial mapping is not constant over the field-of-view (Table 3). The large angle of the stereo cameras, with respect to the z-axis, results in a magnification change over the field-of-view. Hence, there is a significant difference between the vertical and horizontal resolutions, where the horizontal resolution changes approximately 20% over the field-of-view. Furthermore, the accuracy changes as function of the camera separation angle (Lawson and Wu 1997).

A measure of accuracy for the self-calibration procedure is the standard deviation per still water height. Therefore, a selection of still water levels used in the self-calibration procedure for a calibration domain of \(z_\mathrm{d} \approx 62 \, \mathrm {mm}\), measurement plane spacing of \({\varDelta }z \approx 3 \, \mathrm {mm}\), an initial domain size of \({\varDelta }z_\mathrm{d} =80 \, \mathrm {mm}\), and camera separation angle of \(2\alpha = 50^{\circ }\) is shown in Fig. 5a. The standard deviation varies from \(\sigma _{\mathrm {up}} = 0.52 \, \mathrm {mm}\) to \(\sigma _{\mathrm {down}} = 0.30 \, \mathrm {mm}\) over the height \(-110\le y \le 50 \, \mathrm {mm}\) with an average of \({\overline{\sigma }} = 0.38 \, \mathrm {mm}\); see Table 3. The higher standard deviation of \(\sigma _{\mathrm{up}}\) is either caused by the increased magnification at the top of the domain or the limited depth-of-field (i.e., the edge of the focal domain), which is evident from the larger spread over the x-direction (Fig. 5b). The measurement planes are positioned at a small angle, which remains minimal (\(2.3^{\circ }\)) even at the outer edges of the domain (Fig. 5b). Furthermore, the angle could be removed easily by interpolating the data to a vertical plane. The virtual location of the scanning mirror, that is the intersection point of the measurement planes, is located at \(y \approx 1743 \, \mathrm {mm}\) above the false bottom, which corresponds with the measured value of \(y_c \approx 1745 \, \mathrm {mm}\).

The accuracy of the self-calibration procedure is further evaluated by comparing the still water levels reconstructed with the reference and stereo measurement system (Fig. 6). Measurements are performed with the stereo cameras at two different separation angles (\(2\alpha \)), while maintaining a constant focal point. The zero level correction is performed at equal still water level for both the stereo and reference measurement. However, the zero level is not equal for the two separation angles.

The free surface height is reconstructed accurately over the domain size of \(-\,100\le y \le 100 \, \mathrm {mm}\) without significant influence of the domain size (\(z_\mathrm{d}\)) and inter-plane spacing (\({\varDelta }z\)) for both camera separation angles (Fig. 6). The difference between \(y_{{\mathrm {stereo}}} - y_{{\mathrm {ref}}}\) over the domain size, for either separation angle \(2\alpha =30^{\circ }\) (indicated by open squares) or \(2\alpha =50^{\circ }\) (indicated by open circles), is relatively small, as shown in the inset of Fig. 6. For both separation angles, the error remains within 1% of the target domain size (L) of \(100 \, \mathrm {mm}\). A systematic error is observed, which clearly displays a parabolic behavior.

The free surface waves behind an obstacle are measured at two separation angles (\(2\alpha \)) for three different Froude numbers (Table 2). The free surface wave patterns vary over a significant height, which is ideal for a proof of principle measurement. Furthermore, the wave patterns display a variety of length scales from capillary waves to large free surface undulations as shown in Fig. 7. These free surface undulations are clearly observed to travel downstream as shown in the animation (Online Resource 2).

An instantaneous snapshot of the free surface height (y) is shown in Fig. 8a, d, g for the three different Froude numbers. The measurements are obtained for a domain size of \(z_\mathrm{d} \approx 62 \, \mathrm {mm}\), plane spacing of \({\varDelta }z = 3 \, \mathrm {mm}\), and a separation angle of \(2\alpha = 50^{\circ }\). The free surface height is reconstructed over \(N_\mathrm{p} = 19\) measurement planes with an acquisition frequency of \(f_\mathrm{a} = f/N_\mathrm{p} \approx 53 \, \mathrm {Hz}\) per scan, where the cameras and scanning mirror operate at \(f= 1 \, \mathrm {kHz}\).

The differences in amplitude and wavelength of the free surface waves are apparent. The temporal variance of (\(\eta (x,z,t) = y-{\overline{y}}\)), with respect to the temporal mean \({\overline{y}}\), is clearly observed in a space–time diagram. These space–time diagrams are obtained at two locations, spanning the z-direction, as shown in Fig. 8b–f, h, i).

Limited free surface disturbances, without a repetitive pattern, are expected at \(x = -\,40 \, \mathrm {mm}\) (Fig. 8b, e, h), as the upstream flow is relatively uniform. There is indeed no apparent pattern visible over the z-direction. However, the temporal fluctuations (\(y - {\bar{y}}\)) for the first two cases are still significant (\(\sigma _1 = 0.34 \; \mathrm {and} \; \sigma _2 = 0.48 \, \mathrm {mm}\)). These periodic wave shape variations over time (Fig. 10b, e) are introduced by fluctuations in the upstream flow. For \(\text {Fr} =0.094\) (Fig. 10h) the velocity fluctuations have a negligible influence on the drowned hydraulic jump. Therefore, the standard deviation for the measurement at \(\text {Fr} = 0.094\) is considered as a representable error measure, which has a magnitude of \(\sigma _3 \approx 0.15 \, \mathrm {mm}\). The bias error, between the stereo and reference measurement, defined as the difference between the temporal average of the stereo and reference measurement, is \(1.0 \, \mathrm {mm}\) for \(\text {Fr} = 0.094\), or 1% of the target domain size.

The second location is defined for each Froude number, to capture the changes in the free surface behavior. For \(\text {Fr} = 0.044\) at \(x = 20 \, \mathrm {mm}\) (Fig. 8c), small variations in spanwise wave behavior are observed. For the \(\text {Fr} = 0.054\) at \(x = 10 \, \mathrm {mm}\) (Fig. 8f), the spanwise variations are more pronounced with an increased disturbance amplitude. Furthermore, large, spanwise moving, surface disturbances are observed. For the final measurement at \(\text {Fr} = 0.094\), a relatively stationary free surface is expected. However, large free surface disturbances are observed. These disturbances occur due to air bubbles trapped on top of the recirculating flow of the submerged hydraulic jump (Te Chow 1959), which can be clearly observed in the animation (Online Resource 3).

The shape of the crescent-shaped disturbance, with preceding capillary waves, observed in Fig. 8d resembles the instabilities observed on spilling breakers (Duncan et al. 1999; Su et al. 1982). The surface disturbance shown by Gui et al. (2014) is, however, not observed with the current free surface measurements.

It is found that bubbles on the surface seriously deteriorate the signal quality, so that the measurement technique is not suited for flows with a lot of air bubbles. An example of such a detection error is shown in Fig. 9. The viewing lines of stereo-camera 1 cross the interface of an air bubble (Fig. 9a). This is registered as an additional interface above the true interface, due to the reflection of light in the air bubble. Camera 2 registers a shadow below the true interface, as the fluorescent light is reflected (Fig. 9b). However, the error introduced by the air bubble is limited (Fig. 9c). In the current work, a quality indicator (e.g., the vector norm of the distance between triangulated and mapped data) is not included, but more elaborate validation and/or processing can easily be implemented for particular situations (e.g., do not average if one of the cameras has “step-like” results).

A comparison between the temporal mean of the stereo and reference measurements is shown in Fig. 10a. The average difference is negligible (\(0.2 \, \mathrm {mm}\) or 0.2% of the target domain size) with a maximum deviation of \(0.45 \, \mathrm {mm}\). The error for \(\text {Fr} = 0.044\) seems to be considerably larger. However, the standard deviation of the difference is, for both cases (i.e., \(\sigma _1 = 0.45 \; \mathrm {and} \; \sigma _2 = 0.28 \, \mathrm {mm}\)), comparable to the previous error definition. The amplitude of the free surface disturbance is estimated as \(A_1 \approx 3.2 \, \mathrm {mm} \) and \(A_2 \approx 8.6 \, \mathrm {mm}\), which results in a relative error (\(\sigma _i/A_i\)), as defined in Table 1, of 14% and 10% respectively. However, the relative error compared to the target domain size (\(\sigma _i/L\)) is negligible (\(\sim \, 0.4\%\)). For the smaller camera separation angle (\(2\alpha = 30^{\circ }\)), the averaged difference remains the same; whereas the maximum increases to \(0.81 \, \mathrm {mm}\). The reference camera system also has an inherent error, which for a typical PLIF system is approximately \(0.5 \, \mathrm {mm}\) (Duncan et al. 1999). In this case, the typical error of the reference system is estimated as the error induced by a single pixel shift, which corresponds to \(0.1 \, \mathrm {mm}\) or approximately 1–3% of the amplitude. The errors of the stereo-camera system are comparable to typical PLIF systems (e.g., Buckley and Veron 2017; Duncan et al. 1999).

The space–time diagrams of the disturbances (\(\eta (x,0,t) = y-{\overline{y}}\)) are, for \(\text {Fr} = 0.044 \; \mathrm {and} \; \text {Fr} = 0.054\), shown in Fig. 10b, c. These results are obtained at the center line of the domain. The growth and decay of the free surface disturbances are visible. First of all, the amplitude of the free surface disturbances is observed to be larger for \(\text {Fr} = 0.044 \), whereas the occurrence frequency is higher for case one. Second, for \(\text {Fr} = 0.054\) a crescent type of disturbance is observed, where the small amplitude waves are observed to travel upstream (towards negative x).

The local phase speed of the disturbances is, for both cases, estimated with a line as \(c_{p,1} \approx 24.7\) and \(c_{p,2} \approx 29.7 \, {\mathrm {cm~s}}^{-1}\), respectively, (Fig. 10b, c). The estimate is checked by comparing the wavelength of the disturbances on the wave crest for \(\text {Fr} = 0.054\) (Fig. 7a) with linear theory. A zoom-in of the disturbance is shown in Fig. 7b, where peaks (open squares) and troughs (open circles) are indicated. The wavelength \(\lambda \) of the first disturbance, defined as the difference between crests and troughs, is approximately \(\lambda _1 = 4.8 \, \mathrm {mm}\) (\(k = 1.3 \times 10^3 \, {\mathrm {m}}^{-1}\)). The dispersion relation for gravity–capillary waves on still water is,with \(\gamma \) the surface tension, \(k=2\pi /\lambda \) the wavenumber, and H the water depth. For deep-water surface waves, the depth does not influence the dispersion relation (i.e., \(\tanh (kH)\cong 1\)) (Lamb 1993; Whitham 1999). The local phase velocity, on still water, is defined as \(c = \omega /k\). Based on the wavelength of the first disturbance (\(k = 1.3\times 10^{3} \, {\mathrm {m}}^{-1}\)) with surface tension and density values of \(\gamma = 0.072\, {\mathrm {N~m}}^{-1}\), \(\rho = 998 \, {\mathrm {kg~m}}^{-3}\), respectively, the phase velocity is estimated as \(c_{p,\lambda _1} \approx 32 \, {\mathrm {cm~s}}^{-1}\). The wavelength of the following crest is estimated as \(\lambda _2 = 7.3 \, \mathrm {mm}\), which results in a phase velocity of \(c_{p,\lambda _2} \approx 28 \, {\mathrm {cm~s}}^{-1}\). The determined phase velocities are comparable to the estimate of \(c_{p,2} \approx 29.7 \, {\mathrm {cm~s}}^{-1}\) (Fig. 10c). A small difference is expected as the space–time diagram is determined in a fixed frame of reference, whereas the phase velocity is based on a moving frame of reference. The order of magnitude of the Doppler shift \(U = c_{p,2} - {\overline{c}}_{p,\lambda } \approx 2.4 \, {\mathrm {cm~s}}^{-1}\) is comparable with the upstream bulk velocity (Table 2). The ability of the method to measure capillary waves is indicated by the estimated wavelength and corresponding phase velocity.

As a second test case, a droplet impact is generated by releasing a water drop of approximately 5 mm in diameter from a height of 757 mm to impact on a deep-liquid pool (\(H_0 = 141 \, \mathrm {mm}\)). The resulting free surface disturbances are recorded with the stereo PLIF system in exactly the same configuration as in the first test case, with a domain size of \(z_{\mathrm{d}} \approx 62 \, \mathrm {mm}\), a plane spacing of \({\varDelta }z = 3 \, \mathrm {mm}\), and a camera separation angle of \(2\alpha = 50^{\circ }\). The measurements are timed to obtain two impact events in one recording. However, the droplet release time is not synchronized with the recording time.

The free surface disturbances resulting from the droplet impact are clearly observed in the animation (Online Resource 4). Three instantaneous snapshots of the ring waves, generated by the second impact, are shown in Fig. 11a–c. The two markers indicate the locations where time series are extracted, as shown in Fig. 11d.

The first time instance (t1) is approximately \(228 \, \mathrm {ms}\) after the droplet impact at \(\mathbf {X}_p \approx (-\,60.0,0.0) \, \mathrm {mm}\) (Fig. 11a). The maximum free surface disturbance \(\eta (x,z,t) = y-{\bar{y}}\) is limited to \(\pm \,1.5\, \mathrm {mm}\) to remove the unphysical amplitude introduced by interference of either the incoming water drop or the rebounding liquid jet with the light sheet.

At the second time instance (t2), the marker (\(\circ \)) is located in the trough of the first wave train (Fig. 11b). The small amplitude and short wavelength waves preceding the second wave train are also observed. The marker (\(\triangle \)) observes the small wavelength waves preceding the first wave train. Furthermore, the second wave train generated by the droplet rebound is observed in the instantaneous snapshot of the free surface height.

Finally, the third time instance (t3) shows clearly the amplitude decay (Fig. 11c), as energy is conserved, while the front and back group velocities diverge. The amplitude decay is proportional to time as \(\eta \propto t^{-1/2}\) (Whitham 1999).

A space–time diagram displays the free surface variation along the center line (\(z=0 \, \mathrm {mm}\)) of the measurement domain, which is used to estimate the local phase velocity (\(c_\mathrm{p}\)) (Fig. 12). The minimum group velocity (\(c_{g,\mathrm {min}}\)) can, however, not be estimated according to Moisy et al. (2009), as the wave amplitude is below the measurement accuracy of the current stereo-PLIF configuration.

The local phase velocities of the wave crests, resulting from the droplet impact and rebound, are estimated as \(c_\mathrm{p} \approx 27.5 \, {\mathrm {cm~s}}^{-1}\) using the space–time diagram in Fig. 12. The local phase velocity is larger than the minimum phase velocity for surface waves \(c_{p,\mathrm {min}} = (4g\gamma /\rho )^{1/4} = 23.1 \, {\mathrm {cm~s}}^{-1}\) with a wavelength at minimum phase speed of \(\lambda _{\mathrm{p,min}} = 2\pi \sqrt{\gamma /(\rho g)} = 17.3 \, \mathrm {mm}\). The wavelength of the surface waves, determined as the distance between the first peak and trough at \(t_3\) (Fig. 11c), is approximately \(\lambda \approx 44 \, \mathrm {mm}\). The observed waves are deep-water gravity waves (\(\lambda < H_0/2\)) for which the local phase velocity \(c_\mathrm{p} = \sqrt{(g + \gamma \rho ^{-1}k^2)/k} \approx 28.1 \, {\mathrm {cm~s}}^{-1}\) is comparable to the local phase velocity estimated from the space–time diagram shown in Fig. 12.

After, approximately, \(t_\mathrm{r} \approx 2.6 \, \mathrm {s}\) another free surface disturbance occurs (Figs. 11d, 12). This free surface disturbance is induced by the reflection of the initial wave train from the side walls of the channel. The reflected wave traveled a distance of \(z \approx 600 \, \mathrm {mm}\) during \(t_\mathrm{r} \approx 2.6 \, \mathrm {s}\), which corresponds to a local phase velocity of \(c_\mathrm{p} \approx 23 \, {\mathrm {cm~s}}^{-1}\). This local phase velocity is close to the minimum phase velocity (\(c_{\mathrm{p,min}}\)) of surface gravity waves.

A skew is introduced in the two-dimensional free surface measurements (\(\eta (x,z,t)\)), as is clearly observed in Fig. 13a. The scanning system acquires line measurements over the z-coordinate with a temporal spacing of \({\varDelta }t = 1 \, \mathrm {ms}\). In the current example, there are \(N_\mathrm{p} = 19\) light sheet locations. The difference between the first and last measurement plane is significant (\(18 \, \mathrm {ms}\)). Based on the local phase velocity, this introduces a relative displacement of \({\varDelta }x = c_\mathrm{p} ({\varDelta }t N_\mathrm{p}) \approx 5.2 \, \mathrm {mm}\). The measurements are, therefore, non-instantaneous, as the measurement time is too large which introduces a significant displacement.

Temporal interpolation can be applied to reduce the skew (Wellander et al. 2014) (Fig. 13d). The instantaneous free surface height (Fig. 13a) is interpolated to the center plane time (t), where a linear change in amplitude is assumed (Fig. 13b). The absolute difference between original (\(\eta \)) and interpolated (\(\eta _{\mathrm {i}}\)) data shows an amplitude deviation of \(|\eta _{\mathrm {i}} - \eta | \approx 0.3 \, \mathrm {mm}\) with a horizontal shift of \({\varDelta }x \approx 2.5 \, \mathrm {mm}\) (Fig. 13c). The horizontal shift corresponds to the displacement based on the local phase velocity, with the time over half the domain size \({\varDelta }t = N/f = 9/1000 = 9 \, \mathrm {ms}\) and local phase velocity \(c_\mathrm{p} = 27.5 \, {\mathrm {cm~s}}^{-1}\) as defined in Fig. 12.