Plume Spreading Due to Floor Conditions of a Plunging Liquid Jet Using Stereographic Backlit Imaging

Plunging liquid jets are a complex multiphase flow created by the impact of a liquid jet on a liquid pool. The primary focus of plunging liquid jet research is the resultant bubble plume formed beneath the liquid surface created by gas entrainment due to shear effects at the edge of the jet. Plunging liquid jets are researched to understand the high efficiency liquid–gas mixing, which nature uses for aeration from waterfalls, rapids, and breaking waves [1]. For example, plunging liquid jets are responsible for the bulk of O₂ and CO₂ entrained at the free surface of oceans, lakes, rivers, and streams [2]. If adequate control of the aeration and mixing process could be achieved, industrial applications of plunging liquid jets would be ideal for operations such as the mixing of gas–liquid fuels [3,4]. Plunging liquid jets are already used in applications such as dams, fish farms, chemical reactors, nuclear cooling systems, and wastewater treatment facilities [1,4–6].

In a plunging liquid jet system, air can be entrained at the edge of the jet or within the free jet depending upon the flow characteristics. As the liquid jet impacts the liquid surface, a gas film is produced around the perimeter of the jet due to shear forces at the edge of the jet [2]. Air entrainment is attributed to the breakup of these gas films due to instabilities in the plunging liquid jet [2]. This process was examined by Zhu et al. [6] who was able to produce a highly stable jet without air entrainment at flow rates higher than the previously accepted critical entrainment velocity. Low flowrate liquid jets with minimal disturbance will lead to sporadic entrainment [2]. Further increasing the liquid flowrate leads to continuous entrainment from a constantly collapsing pocket formed around the incoming liquid jet [2]. This transition takes place somewhere between a liquid Reynolds number at the jet exit of 3500 ≤ Re_N ≤ 4500; an example of these two entrainment regimes is shown in Fig. 1.

Of the many variables adjusted to experimentally investigate the bubble plume from plunging liquid jets, the current study focuses on the effects of bubble plume spread. In an unobstructed vertical plunging liquid jet, the rising bubbles are often re-entrained as they rise in opposition to the incoming fluid flow. The influence of these rising bubbles is observed in experiments where the incoming liquid jet was angled or oscillated to separate the incoming flow from the bubble plume, resulting in a significant increase in bubble plume penetration depth [7]. An example of how the separation of rising bubbles from the incoming plume can result in increased penetration depth using the current flow system described below is shown in Fig. 2.

While the inclusion of a tank floor does not allow for a change in the penetration depth, it does spread the bubble plume, reducing the resistance to the incoming plunging liquid jet. The bubble plume–floor interaction has not been significantly explored within the literature, but is found in many real-world applications, such as waterfalls, breaking waves on the beach, filling of bottles, and molten metal casting, to name a few. The focus of this study is to expand a model for bubble plume width in an infinite pool depth to one with a finite pool depth, and then compare the expanded model to relevant experimental data.

Simple theoretical models have been proposed by various authors trying to explain the bubble plume maximum penetration depth, plume width, entrainment rate, and other properties of a simple plunging liquid jet [3,8,9]. This study focuses on entrainment from a single cylindrical small-scale plunging liquid jet with a circular cross section. Infinite depth planar or “2D” plunging liquid jets have also been studied [1,4,10,11], but are not the focus of this work.

Both Refs. [8] and [9] further assume the bubble plume expands at a fixed rate before the momentum is overcome and the bubbles begin to travel back to the surface. The control volume used for this infinite depth bubble plume is shown in Fig. 3(a).

Note that both authors also assumed that the diameter of the plunging liquid jet entering the control volume was approximately the same as that of the nozzle exit and that the void fraction approaches zero at z₁.

This value inherits the assumption made in Eq. (6), that D₁ ≫ D₀ = D_N.

where v₁ and ϵ₁ are replaced by v_max and ϵ_max, representing the velocity and void fraction at the new control volume exit location. Note, however, that D₁ is not directly replaced with D_max because the exit of the control volume is now a ring with outside diameter D_max and inside diameter D′. Equation (9) represents bubble plumes for which the base position is above the maximum width location z₁ when there is no base plate. Otherwise, portions of the plume would already have left the defined control volume.

To solve Eq. (11), bubble rise velocity at the plume exit (v_max) and the inside diameter of the exiting plume (D′) need to be further defined. The bubble rise velocity at the control volume exit cannot be easily defined by buoyancy like it was in the infinite depth system [9]. Without a physical measurement of the bubble rise velocity, however, the best approximation available is to assume that v_max reaches the terminal bubble rise velocity (v_t) when the bubble reaches the maximum width (v_max ≈ v_t). This means that the first term under the square root in Eq. (11) is D₁² of the infinite depth system (see Eq. (8)). Even if v_max ≠ v_t, so long as the two values are constant, the result should be proportional to D₁².

The point at which Eq. (12) transitions from the infinite depth case to the finite depth case is somewhere between z₁ and Z. As shown in Fig. 3(a), at z₁, the flow exits the defined control volume, meaning that the entire flow does not impact the plate.

An approximation of the location of the maximum width is noticeably absent from this method of solving Eq. (9) unlike that presented for the infinite depth jet [8,9]. While the growth of the bubble plume has previously been shown to expand at a constant half angle (α), allowing for simplification using Eq. (5), neither the position above the baseplate (t in Fig. 3(d)) nor the angle at which the bubble plume expands after impact (α′ in Fig. 3(d)) has been explored by existing literature. Since the primary focus of this study is the growth of the width, only D_max and D′ are explored here.

Note that while Eqs. (7), (8), and (12) could all be easily converted to nondimensional Reynolds numbers because v_ND_N is used in the nozzle Reynolds number (Re_N), and the bubble terminal velocity, v_t, is also present. The characteristic length of this Reynolds number would be the bubble diameter, which is varied and unknown. Furthermore, the existing standard for current plunging liquid jet publications leans heavily toward the use of v_ND_N for plotting and assessing trends in plunging liquid jet systems [3,9,12]. Hence, the dimensional forms will be used below, and jet velocity is provided in terms of v_ND_N.

A 32 cm square acrylic tank with an adjustable floor plate was used to simulate the presence of floor interactions in a plunging liquid jet system. Flow was introduced and controlled using a pressurized air system to form a continuous fluid stream through a 2.1 mm nozzle schematically detailed in Fig. 4. The liquid nozzle length was 110 mm, providing a length-to-diameter ratio of 52, ensuring a fully developed flow at the exit. The water level within the imaging tank was maintained through a secondary tank with an overflow weir connected through the base of the tank. This minimized the change in liquid depth, which occurred only at high liquid flow rates and is described in detail elsewhere [13].

Imaging of this system was completed using backlit imaging and two perpendicular high-intensity LED panels whose light emissions were in the red and green spectra, respectively. These were paired with two Photron SA-Z cameras with 180 mm Nikon lenses positioned opposite the two perpendicular LED panels and equipped with red and green filters which removed over 99% of the perpendicular light source while allowing over 90% transmission of the opposing backlight. Resulting images had an approximate resolution of 8.6 pixels per millimeter with a total image size of 11.9 cm by 11.9 cm, which was further reduced to 11.4 cm wide by 7.5 cm tall due to the size of the uniform light sources as well as other restrictions fully described in Ref. [13].

Each camera captured sets of 1000 images at 250 frames per second (fps) for sixteen liquid flow rates (from Re_n ≈ 2750 to Re_n ≈ 16,600) and seven different tank base positions (not including the infinite depth reference condition). While faster capture rates were possible with this experimental setup, initial trials indicated that a longer capture length was required to capture average trends. Averaged values for penetration depth, plume width, and projected area were seen to provide consistent results between trials when averaged over 1–4 s intervals [13]. Data were also taken at alternating ascending and descending flow rates to eliminate any bias, with ten repetitions per flowrate and base position combination. A full summary of the accuracy and variation in flow conditions assessed in this experiment is provided in Ref. [13].

Individual images were computer processed using consistent controls detailed in Ref. [13]. In summary, images were first normalized through comparison to no-flow conditions to remove artifacts from the tank and camera. Next, bubble identification was completed based on a quantitatively determined, but consistent threshold value. Further noise reduction was completed through a morphological opening procedure. Finally, bright bubble centers caused by bubble lensing effects were removed based on size as well as identified single bubbles smaller than the set noise threshold. These steps are further detailed and optimized in Ref. [13]. For the infinite depth liquid jet system, bubble depth measurements were consistent between paired stereographic backlit images. In the infinite depth configuration, the captured bubble plume depth of the individual images was defined as the depth reached by the lowest identified bubble position in each image and following Ref. [13], only bubbles with a projected area greater than 0.41 mm² were recorded. This value is identified as the “reduced depth.”

The analysis from Ref. [13] also provides other measures including the projected bubble plume area and width. These values were measured based on the projected light path across the whole volume from each of the two perpendicular camera views. Figure 5 provides a visual summary of the measurements taken in Ref. [13] to evaluate the measured bubble plume penetration depth and maximum bubble plume width used in this study for model comparison. An automated identification was completed on the instantaneous images (Fig. 5(a)) to identify the bubble region of the image as shown in Fig. 5(b). The “full” bubble plume was identified as the bubble region with the bubbles smaller than 0.10 mm² (seven pixels) removed for noise control. This provides the best qualitative match to the original images; however, the “full width” and “full depth” measures were too sporadic because of the presence of small bubbles, which are not normally considered in measured bubble plume characteristics. Instead “reduced” images were used for instantaneous “reduced depth” and “reduced width” calculations, which removed bubbles with a projected area smaller than 0.41 mm². As summarized in Ref. [13], the 0.41 mm² projected area (30 pixels) was selected to remove the smallest identified bubbles because they were sporadic and did not adequately represent general bubble plume traits. Note that as shown in Fig. 5(b), the width calculations represent the maximum bubble plume width at any given depth, not the continuous width of the bubble plume. These calculations were evaluated both for temporal variation (Fig. 5(c)) and for average properties (Fig. 5(d)).

Alternatively, the instantaneous images can be averaged to create an average projected plume density contour image, reflecting the likelihood of bubble presence at any given coordinate of the image as shown in Fig. 5(e). Each given contour percentage represents the contour in which the location has a bubble present in X% of the instantaneous images. Note that this is not a void fraction, as backlit imaging only provides information if a bubble is present between the light source and camera but no information on how many. This method provides a reasonable estimate for the average projected bubble plume shape. For this process, the full image was used for the best accuracy. The bubble plume width and depth could then be easily evaluated for any bubble presence likelihood contour as shown in Fig. 5(f). The 60% contour was used for this process because it best qualitatively represented the bubble plume when compared to the original images [13]. Note, however, that other contours would represent similar trends but with a slight offset of measured values. Using these images for bubble plume measurements has the advantage of removing sporadic bubbles and providing a smoother, more defined, and consistent bubble plume shape. Note that other methods of analysis were explored in Ref. [13].

While the average shape of an infinite depth plunging liquid jet bubble plume is near teardrop, as used for the theoretical assessment, the instantaneous jet takes a much more irregular shape (see Fig. 5(b), for example,) which makes it difficult to quantify instantaneous gas entrainment. Figure 6 shows the instantaneous maximum reduced width from the pair of stereographic projections. The instantaneous bubble plume width is not axially symmetric, and the variations between the two images are not correlated. This was confirmed by a fast Fourier transform on the data, revealing random fluctuations [13]. Potential fluctuations in the liquid jet velocity could cause variations in bubble plume width, but as shown in Fig. 6, there is no visible correlation with measured liquid flowrate, which varies over the time span by less than 0.2%, with intermittent fluctuations much less than this. In contrast, the maximum bubble plume depth does match between the two projected images [13]. Also note that while instantaneous results are not axial symmetric, when measured as an average projected bubble plume density as shown in Figs. 5(e) and 5(f), the projections from each camera provide similar measurements [13].

where C₁ is a constant. When C₁ = 0, Eq. (13) refers to the position of the maximum diameter, z₁, from Eq. (6). When C₁ = 0.5, the base of the bubble plume is modeled as hemispherical, as assumed in Eq. (7) and by Ref. [8]. If C₁ > 0.5, the bubble plume is ellipsoidal as observed experimentally.

where D₁ and z₁ were calculated experimentally, and D₀ ≈ D_N. Other publications have also approximate D₀ ≈ 0 when determining α [8]. Using Eq. (14), the bubble plume half angle measurements in this study range between 9 and 13 deg, with larger and more varied results for sporadic entrainment cases at the lower liquid flow rates.

The primary interest in tracking the infinite depth bubble plume depth was in the comparison of this system with current literature as a baseline for when the bubble plume would impact the base plate. To solve Eq. (7) for depth, a bubble terminal rise velocity value of v_t = 0.22 m/s was used, following Clanet and Lasheras [8]. Using this value, it is possible to solve Eq. (7) for the theoretical bubble plume depth as a function of the nozzle exit velocity, nozzle diameter, and bubble plume half angle.

Figure 7 shows the theoretical values in comparison to the average reduced bubble plume depth from instantaneous images (“+” and “x” symbols on the graph), as well as the 60% contour of the average projected bubble plume density (triangle and circle symbols on the graph). Infinite depth control data were taken both at the beginning and the end of the experiment as the tank required adjustment for smaller increments of plate movement, as described in Ref. [13]. The data collected is highly repeatable for the continuous entrainment regime (v_ND_N > 0.004 m²/s), including between the two discreet capture sets. The values compare favorably when compared to Eq. (7) with α = 12.50 deg, which represents the theoretical model as proposed by Ref. [8].

To evaluate discrepancies in the shape of the bubble plume, as detailed previously, Eq. (13) was used to evaluate the shape of the base of the bubble plume. The hemispherical assumption made by Ref. [8] correlates with C₁ = 0.5, which is the same as Eq. (7). The use of an ellipsoidal shape with 1 ≤ C₁ ≤ 1.5, as found in experimental analysis, significantly overestimates the bubble plume penetration depth, Z, as shown Fig. 7. Alternatively, setting C₁ = 0.5 assumes a hemispherical cone for the depth beyond z₁ from Fig. 3, resulting in a closer correlation with the 60% contour from the average projected bubble plume density. These comparisons indicate that while the qualitative shape of the bubble plume is not as well defined by the theoretical model, it does provide a closer quantitative match for the penetration depth.

One possible reason for this discrepancy is that multiple assumptions were made that would decrease the penetration depth of the bubble plume. For example, buoyant forces, resistance from rising bubbles, and shear forces at the edge of the bubble plume were not considered in Eq. (7). Additionally, Eq. (7) also relies on only taking into consideration the downward moving bubbles. The rising bubbles on the outside of the bubble plume, which would increase the width and decrease the depth at which this maximum width was located, were not considered. As backlit imaging cannot distinguish between the two bubble plume flow directions, the theoretical shape cannot be entirely dismissed.

As noted by Refs. [8] and [9], the bubble plume penetration depth as a function of v_ND_N is near linear, indicating that the bubble plume half angle is near constant. When α = 12.50 deg, the slope of the linear relationship is approximately 12.5 (and α = 11.80 deg –12.80 deg results in a range of slopes of 13.2–12.3). Equation (7) is also plotted using the instantaneous experimental half angles, calculated using measured z₁ and D₁ in Eq. (14), but does not improve the overall fit of the data.

Published experimental results for the bubble plume penetration depth as a function of v_ND_N have found the slope of this linear function to be 10 s/m [9] and 10.25 s/m [8], which correspond well with the results here. A best-fit linear relationship of the average projected bubble plume density produced a slope of 12.13 s/m, labeled “Eq. (21)” in Fig. 8. As shown in Table 1 and displayed in Fig. 8, not all trends presented in the literature are linear, but are near-linear in the liquid flow ranges studied here. Note that the liquid flowrate range for the current study was limited to where the bubble plume remained in the image region cast by the LED backlight. This region does not extend past where Eq. (17) predicts a significant change in slope, though Eq. (21) is used later as an approximation for Z in Eq. (12).

While the trends match those available in the literature, the equations are discrete. These differences can be minimized through variation in the choice of threshold values for image identification. While only a selection of published experimental fits is provided here, there is a noticeable variation between experiments. This is caused by varying definitions for identifying plume features (thresholds) as well as variation in the instabilities in the experimental systems. For a full explanation of the image analysis methods used for the current data, see Ref. [13]. Matching trends observed by others for an infinite depth bubble plume is important to verify image analysis techniques; these validated techniques are then used to investigate plume spreading due to tank base interactions to which no comparisons are available in the literature.

To assess the impact of the false floor, a baseline bubble plume width (D₁) is required for comparison with published data and for use within Eq. (12) to determine the finite depth maximum width (D_max). Following Eq. (8), the maximum bubble plume width, D₁, should be linearly correlated to v_ND_N with a slope of 1/v_t. As stated above, v_t ≈ 0.22 m/s is the value used by [8] for this approximation. As shown in Fig. 9, Eq. (8) follows the experimental trends for the average instantaneous bubble plume characteristics but underestimates the values. This is not surprising, as the instantaneous bubble plume includes large rising bubbles at the edge of the bubble plume. In the same way, Eq. (8) overestimates the values for the 60% contour of the average projected bubble plume density data. This is because this method rules out any area in which bubbles are not present at least 60% of the time, creating a symmetrical, but reduced estimation of the bubble plume shape. For later use in Eq. (12), the best fit for the 60% contour of the average projected bubble plume density was used. Figure 9 also includes the best fit values for other contour choices for reference.

Figure 10 shows the effect of base plate position on the maximum bubble plume width, determined by the 60% contour line from the average projected image. Note that the data points displayed here represent the average of all the width values from all ten replicates of both stereographic cameras. The location when the bubble plume hits the base plate at its respective position, as estimated by Eq. (21), is also identified. When comparing the growth in width after hitting the tank floor, there is a slight delay before the width begins to increase at a near linear rate. This has to do with the distance between z₁ and Z where the infinite depth bubble plume is a hemisphere (as shown in Fig. 3). In this region, the edge of the bubble plume has already left the defined control volume before contact with the base plate. Equation (12) does not account for this transition, leading to the delayed behavior.

The growth in maximum width after the impact of the plate itself, as shown in Fig. 10, increases in a shallow quadratic fashion with an asymptote somewhere outside of the range of studied flow rates. The exception to this is the h = 23 mm plate position which appears linear. The h = 23 mm position was also the most difficult to capture as the plate partially obscured the projected image to the camera because the camera location was fixed for all conditions, and when h = 23 mm, the plate is slightly above the camera center. It is believed this caused the difference in the observed trends.

To solve Eq. (22), values for Z and D₁ were pulled from the best fit lines presented in Figs. 8 and 9. Note that these fits were based on trends in the experimental data which could be collected within the backlit imaging region. This meant that only flow rates v_ND_N < 0.008 and 0.010 were used to set trends for Z and D₁ approximations, respectively. As shown in Fig. 11, Eq. (22) reasonably matches the experimental data. However, a single correlation does not perfectly match every plate position. By varying the chosen constants slightly, a better equation could be found for each of the individual plate positions. While this could be due to slight depth and measurement fluctuations, the trend appears to indicate there may be an additional impact of the tank depth not covered within this model.

In order to isolate the consistent growth of the bubble plume width after plate impact, the increase in maximum width (D_max – D₁) was compared with the decrease in the depth of the bubble plume (Z – d). These data, shown in Fig. 12, align well across all base positions, suggesting that the change in maximum width is likely a function of the loss in plate depth. While there is slight variation between base positions, there is no general trend with base position. This indicates that the offsets noted in Fig. 11 are either not present in the gain in width (D_max – D₁) to loss in depth (Z – d) comparison, or that there is error associated with the measurement of the base positions.

When comparing the empirical model presented in Eq. (23) to the experimental data in Fig. 13, the overall fit again matches the data very well. However, just as in Fig. 11, some variation exists (particularly when h = 49 mm and 58 mm). Modifying the chosen constants will correct for this change for any given base position, but no single set of constants perfectly defines all base positions.

A similar modeling to that presented in Fig. 12 was completed for (D²_max – D₀²) and ((Z – d)²) resulting in the same form as Eq. (22). Additionally, a comparison (D²_max – D₀²) and (Z*(Z – d)) was also attempted as an alternate interpretation of how to define D′ to that presented in Eq. (12). The results of both models are similar to Fig. 12, and do not correct the fit variation present in Figs. 11 or 13.

Noting there is a relationship between the increase in maximum bubble plume width (D_max – D₁) and loss in depth (Z – d), there may be a relationship between the total bubble plume volume in the infinite plate depth bubble plume compared to the finite plate depth bubble plume due to plate interactions. To prove a connection, a method that can accurately capture the entire bubble plume is required. The data used here is based on the projected bubble plume, which is unable to capture potential voids within the bubble plume area as shown in Fig. 14. While this methodology does capture the maximum width and depth of the bubble plume, it does not capture the volume, void fraction, or inner shape of the bubble plume. However, as a proof of concept, bubble plume volume can still be estimated by assuming a symmetric average bubble plume shape as it does provide a baseline comparison of the spatial impact of the plunging liquid jet and base plate.

The bubble plume volume was estimated by assuming symmetry for the 60% contour of the average projected bubble plume density. The diameter was calculated at each vertical position and revolved around its center to create an estimate of the volume for each dataset. These estimated volumes are displayed in Fig. 15 for both the infinite depth bubble plume (approximately 40 data points (circles) per flowrate) and finite bubble plume (approximately 140 data points (dots) per flowrate). Using this methodology, the volume of the bubble plume appears to have negligible change with changing floor effects as many of the infinite depth and finite depth data points align.

The lines on the graph represent the theoretical volume of the bubble plume based on the teardrop shape displayed in Fig. 3(a) (assuming symmetrically revolved). “theoretical volume 1” was calculated using Eqs. (7) and (8) to calculate Z and D₁, respectively, and, following Ref. [8] α = 12.50 deg and v_t = 0.22 m/s. These assumptions result in a curve that significantly overestimates the volume compared to the calculated data, similar to what was observed in Figs. 7 and 9. This is because the 60% contour of the average projected bubble plume density only represents the region where bubbles are present 60% of the time. This method does create a consistent and symmetrical shape but does so by shrinking the evaluated area. Alternatively, the same teardrop shape was calculated using the experimental trends for Z and D₁ (“theoretical volume 2”), which underestimates the volume. This is owing to the bubble plume shape not truly matching the teardrop shape with a hemispherical base as described in Fig. 3.

This study has thus far provided several trends and observations based on experimental data, along with briefly highlighting the limitations of the methodology used to acquire such data. To provide confidence in the observations recorded here, three uncertainties are detailed below; system uncertainty having to do with the control and variation of the experimental methods, image analysis uncertainty having to do with the limitations and approximations made in the analysis of the data, and trend uncertainty having to do with the equations presented here as well as their forms and constants.

Within the data collection, the primary areas for error include possible variations in the liquid jet flowrate, possible angling of the liquid jet, and the potentially uneven base plate. The relative variation of the flowrate measured by the Altrato flowmeter remained within 0.004 LPM (v_ND_N = 4 × 10^–5 m²/s) of the desired flowrate during data collection, although the accuracy of the flowmeter is only certified to 1% of the measured value (±0.015 LPM or v_ND_N ±1.5 × 10^–4 m²/s at the highest flowrate tested) [13]. While this variation exists within the data collection, flow rates were tested at 0.05 LPM increments (or v_ND_N = 5.1 × 10^–4 m²/s), which is significantly larger than any variation within the flow and therefore should not have a significant impact on the observed trends.

The impact of the possible angling of the liquid jet and potentially uneven base plate on the given trends is more difficult to understand. Both parameters were leveled to within 2 deg of vertical and horizontal, respectively [13]. The influence of the liquid jet angle has been shown to increase the bubble plume penetration depth by 30%–40% for angles around 10 deg [7,8]. However, based on reported data from Ref. [8], experimental trends only appear to increase the depth by ∼15% at 2.5 deg. In the current study, however, the liquid nozzle was not moved throughout data collection. Therefore, while the actual bubble plume penetration depth may be affected, the trends should remain consistent. The impact of an uneven base plate has not been addressed in the literature and would require further studies for proper assessment.

Bubble plume width and depth measurements were completed based on the relative image intensity of individual images of the system through a process fully tested and documented elsewhere [13] (an abbreviated explanation is available within the Experimental Methods section). The primary image analysis uncertainties come from the qualitatively selected thresholding values used in the analysis. For an individual image identification, ideal identification could be set from a range of values which all provided reasonable qualitative results. Within this range of potential threshold values, only 12% of the collected images varied more than 5% from the measures maximum bubble plume width [13].

The use of individual image identification values was impractical due to the large time variation of the bubble plume, even between consecutive images. Over the 20,000+ images captured for each continuous entrainment regime liquid flowrate, the maximum variation in maximum bubble plume width for the instantaneous images was rarely more than 40 mm, while the lowest liquid flow rates produced the sporadic entrainment regime with a larger variation in maximum bubble plume width for the instantaneous images of around 80 mm [13].

While instantaneous variation was found to be random in nature, the average projected bubble plume shape was found to be consistent when each group of consecutively captured images was averaged [13]. The data presented in the current study represents the average projected bubble plume density, which was collected individually over each of the 10 discrete trial sets (each captured from two perpendicular projections). The bubble plume shape was defined by the projected image area in which a bubble was identified in more than 60% of the 1000 captured images over a 4 s image acquisition period (the 60% contour line in Figs. 5(e) and 5(f)). Between the 10 4-sec trials for each text condition, the maximum projected bubble plume width varied no more than 4 mm for all test conditions [13].

The trends in bubble plume width and depth presented here represent a slightly different trend than those in previous publications [3,8,9,16]. Variations may be attributed to the unique system used in this study, as previous publications have noted the variations in published trends based on the differing instabilities of each system [6]. Therefore, the focus of this analysis is on the variation and uncertainty of these trends best matching this study, in that while absolute variations in publications exist, observed general trends tend to be similar [3].

Both the comparison to existing literature as well as the definition of a finite depth bubble plume in this study are dependent upon the trends for infinite depth bubble plumes found in Figs. 7 and 9, which define Z and D₁, respectively. These best-fit lines are for the continuous entrainment regime when, for this study, v_ND_N ≳ 0.004 m²/s or Re_n ≳ 4,400; below this flowrate, bubble entrainment is in the sporadic regime. Equation (21) defines the depth Z of the infinite depth system and is based on data collected between 0.0045 m²/s ≤ v_ND_N ≤ 0.0080 m²/s where the flow is both continuously entraining and within the captured backlit image. The slope of this data is 12.13 s/m but could vary from 11.86 to 13.13 s/m (based on the data acquired). Similarly, the best-fit line for increase in width in an infinite depth bubble plume was calculated between 0.0045 m²/s ≤ v_ND_N ≤ 0.0100 m²/s, resulting in a slope of 2.47 s/m as shown in Fig. 9 (60% contour best-fit line) but the slope could range anywhere from 2.09 to 3.22 s/m based on the acquired data.

When looking at the uncertainty of the data collected and trends observed for the finite depth bubble plume, there are several regions of interest as best explained schematically. Figure 16 shows three sample data sets for a finite depth bubble plume in comparison to the infinite depth bubble plume. The lines drawn represent the average trend while individual data points are also shown. The plot for Fig. 16 is broken into four regions based on the point when the average projected bubble plume hits the base plate. The statistical p-value, as defined by Ref. [17], was used to show differences between data from infinite depth and finite depth bubble plume widths at any given flowrate, v_ND_N.

From Fig. 16, “region 4” is used as a baseline to show that after impact with the base plate, growth in bubble plume width was statistically different from its behavior before impact, justifying the two parts of Eqs. (12), (22), and (23). This region is defined as the point where all data sets for the finite depth plate location are statistically different from the control case (i.e., d = ∞). Statistical difference was defined for this study based on p-value comparisons between the data taken for each condition. Flow rates for this study were taken at increments of v_ND_N ≈ 0.0005, limiting the accuracy at which this transition occurs. However, p-values for all data sets have dropped below 2 × 10^–8 within v_ND_N = 0.0015 of the predicted collision flowrate with some conditions hitting this mark considerably sooner. The predicted collision flowrate was determined based on the 60% contour of the projected plume width best fit line, shown in Fig. 9. Using the same test, the data were shown to be discrete from all other base position data within this region, with only the 49 and 58 mm data sets more similar than a p-value of 1 × 10^–8 (p-value of ∼1 × 10^–4 at v_ND_N = 0.0012 after projected collision with the 49 mm base dropping to ∼3 × 10^–5 at v_ND_N = 0.00127 after projected collision).

“Region 3” in Fig. 16 focuses on what happens just after bubble plume impact with the base plate. Based on the shape of the plume defined in Fig. 3(a), there is some region after impact where the hemispherical/ellipsoidal region impacts the base rather than the conical region. A portion of the bubble plume has already begun to rise before base impact in the region z₁ ≤ d ≤ Z resulting in a smaller locational width in this region, which is why this portion is noticeably absent from the theoretical model in Eq. (12). The data point just after projected collision (v_ND_N < 0.0005 after collision) resulted in an inconclusive comparison with the infinite depth bubble plume condition, with p-values ranging from 0.002 to 0.2 (excluding data within the sporadic entrainment regime). Toward the end of this region 3, as the plate impacts higher on the theoretical hemisphere and the width approaches D₁, p-values decrease reaching p-values of 2 × 10^–3 to 1 × 10^–26 for v_ND_N between 0.0007 and 0.0010 after collision. With this it is difficult to assess the impact, or if there is an impact of the base on the bubble plume when z₁ ≤ d ≤ Z, owing to the relative difference compared to the control data.

“Region 2” is intended to assess the impact of compression effects on the bubble plume without direct impact of the plume on the base plate through p-values taken just before the impact of the base plate. Values in this region range from 0.12 to 0.76, for v_ND_N at collision to 0.0003 after collision (again neglecting two conditions in the sporadic entrainment regime), indicating that effect of the base plate on the bubble plume width before impact is also inconclusive. Note that this region is most likely influenced by the contour line selected in the projected bubble plume average images (60% in this study). As displayed in Fig. 17, where the difference between the actual instantaneous bubble plume and the model bubble plume is highlighted, this does not mean that no bubbles reach the base plate. In practice, this value more closely represents when only the tip of the bubble plume impacts the base plate, however, lower flow rates (“region 1”) are even less likely to be statistically affected by the base plate than this region.

As discussed above, due to variation in the data, no unified set of constants can be statistically presented for the prediction of the width of a finite depth bubble plume across varying pool depths. For the same reason, neither Eqs. (22) nor (23) can be shown to be statistically better than the other within the range of captured images. However, within the experimental range, both methods provide good predictions for the data.

Plunging liquid jets are primarily studied for the highly efficient mixing capabilities of their entrained bubble plumes. While many aspects of the plunging liquid jet bubble plume have been studied, the impact of compression effects caused by a tank base is unknown, even though other bubble plume spreading effects have been shown to influence the resulting bubble plume.

This study explores the relative spreading of the bubble plume resulting from a plunging liquid jet in terms of maximum measured bubble plume width, D_max. The focus of this study is on the continuous entrainment regime, where a quasi-stable time-averaged bubble plume is observed. Experimental results for the bubble plume width and depth of an infinite depth bubble plume correlated well with existing theoretical models. Two new models to predict the impact of the maximum width after impact with a finite tank base were proposed, one based on extension of the infinite depth model (Eq. (22)) and one purely empirical based on physical arguments (Eq. (23)).

These models, which have not been previously explored in literature, provide results that can be justified using existing theoretical trends in published literature and provide a means of predicting plume expansion due to base interactions. While both models reasonably match the experimental data, no single set of constants perfectly matches all base plate depth positions. The small differences could be due to measurement errors or further dependence upon variables not represented within the simplified theoretical framework. Regardless, a strong relationship was shown between the decrease in bubble plume depth due to base interference and the resultant increase in bubble plume width which appears to be independent of base depth.

To fully validate the proposed model, further data are required, particularly to confirm the trends just before and just after bubble plume impact with the tank base. Additionally, higher liquid flow rates and a wider image capture region are required to fully evaluate the exact trends in the growth of the bubble plume width after base impact. Ideally, a better theoretical model for the shape of the bubble plume, not just the maximum width and depth, could be established using an alternate three-dimensional imaging technique to fully evaluate the relative change in instantaneous entrainment between the two models.

Finally, research in this area should lead to understanding the relative impact of the resistance caused by the rising bubble plume as well as the impact of compression effects on the bubble hold-under, mixing, and entrainment of the bubble plume from a plunging liquid jet. This information is valuable for creating and testing models of plunging liquid jets and further understanding and controlling the mixing properties of plunging liquid jets.

The assistance of Mr. Orion Roberts with some of the image analysis is gratefully acknowledged.

• Iowa State University (Bergles Professorship in Thermal Sciences; Funder ID: 10.13039/100009227).

• Office of Naval Research (Grant Nos. N00014-18-1-2319 and N00014-18-1-2380; Funder ID: 10.13039/100000006).

A_0/1 =

area in or out of control volume

d =

floor depth of baseplate from the liquid surface

D₀ =

jet diameter at surface of liquid pool

D₁ =

maximum bubble plume diameter of an infinite depth bubble plume

D_max =

maximum bubble plume diameter of a finite depth bubble plume

D_N =

inside diameter of liquid nozzle used for plunging liquid jet

D =

diameter associated with the inside width of the bubble plume at the exit position of the control volume after baseplate impact

F_cv =

forces within the control volume

h =

free jet length before impacting pool surface

Q =

volumetric flow rate (air)

Re_N =

liquid Reynolds number at the nozzle exit

t =

distance from the base plate to the exit of the jet control volume for a finite depth bubble plume

v_max =

fluid flow velocity at the exit of the control volume for a finite depth bubble plume

v_N =

fluid flow velocity at the nozzle exit

v_t =

terminal bubble rise velocity

v₀ =

fluid flow velocity at liquid surface or velocity into the control volume

v₁ =

fluid flow velocity at maximum diameter of an infinite depth bubble plume or velocity out of the control volume

Z =

infinite depth bubble plume penetration depth

z₁ =

depth of bubble plume expansion region before reaching maximum width

α =

constant half angle growth of plume width below surface

α' =

rebound angle off base plate for a finite depth bubble plume

Δt =

rise distance of bubble plume before exit point of rising bubbles before exiting control volume

Δz =

difference between infinite depth bubble plume depth (Z) and plate depth (d) when the bubble plume impacts the base plate

ϵ₀ =

void fraction at exit of control volume

ϵ₁ =

void fraction at entrance of control volume

ρ_w =

density of the liquid in question (water for data provided)

ρ_0/1 =

density in or out of control volume