2 Analysis

Both here and in WF, $t$ is a monotonically increasing function of eddy size, so the relevant $t$ value is the turnover time of the smallest allowed eddies, whose size is denoted $l_{e}$ . It is assumed that $l_{e}$ approximates the size of droplets formed by such eddies.

The Rayleigh criterion $t_{s}\sim (\unicode[STIX]{x1D70C}l_{d}^{3}/\unicode[STIX]{x1D70E})^{1/2}$ , where $\unicode[STIX]{x1D70C}$ is the liquid density and $\unicode[STIX]{x1D70E}$ is the surface tension, determines the minimum size $l_{d}$ of droplets that can be generated by a given shear $1/t_{s}$ . Substituting $l_{e}$ for $l_{d}$ and $t$ for $t_{s}$ based on assumptions thus far,

is obtained, establishing a relationship between $t$ and the smallest possible eddy size, $l_{e}$ , which is not yet known. To evaluate $l_{e}$ , modelling is used to obtain another relationship between $t$ and $l_{e}$ .

For this purpose, WF use HIT phenomenology. To obtain their breakup-onset scaling, the inertial-range scaling of $t$ is expressed as $t\sim (D/U)(l_{e}/D)^{2/3}$ , where $U$ is the jet bulk velocity. This is equivalent to the specialization of equation (2) of WF to the inertial range. Substitution of this into (2.1) gives $l_{e}/D\sim We^{-3/5}$ , where $We=\unicode[STIX]{x1D70C}U^{2}D/\unicode[STIX]{x1D70E}$ . From this, the inertial-range scaling of $t$ is used to obtain $t\sim (D/U)We^{-2/5}$ .

Based on the assumption in § 1 that the eddy turnover time $t$ approximates the elapsed time until breakup onset, the onset location $x_{i}$ scales as $Ut$ , so

The WF result for $t$ then yields $x_{i}/D\sim We^{-2/5}$ . The assumption that $l_{e}$ approximates the size $l_{d}$ of droplets released at onset then gives $l_{d}/D\sim We^{-3/5}$ .

The analysis based on the BL picture differs from the foregoing only in the modelling that determines the $l_{e}$ dependence of $t$ . The first step is to evaluate the shear at the nozzle wall using the Blasius friction law $2(u_{\unicode[STIX]{x1D70F}}/U)^{2}=CRe^{-1/4}$ (Schlichting & Gersten Reference Schlichting and Gersten2000), where $u_{\unicode[STIX]{x1D70F}}$ is the friction velocity, $Re=UD/\unicode[STIX]{x1D708}$ , $\unicode[STIX]{x1D708}$ is the kinematic viscosity, and the empirical coefficient $C=0.079$ and other numerical factors are henceforth omitted from scaling estimates. Substitution of the definition of $Re$ and the definition $u_{\unicode[STIX]{x1D70F}}^{2}=\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y)$ , where $u(y)$ is the streamwise mean velocity profile and its partial derivative is evaluated at the wall, into the friction law gives the wall-shear scaling $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y\sim (U^{2}/\unicode[STIX]{x1D708})Re^{-1/4}$ and thus

The mean shear is derived as a function of $y$ using the law of the wall, expressed as $u(y)/u_{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D705}^{-1}\ln (yu_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708})+B$ , where $B$ and the von Kármán constant $\unicode[STIX]{x1D705}$ are empirical coefficients. From this, $\text{d}u/\text{d}y\sim u_{\unicode[STIX]{x1D70F}}/y$ , corresponding to the time scale $t_{s}\sim y/u_{\unicode[STIX]{x1D70F}}$ .

It is assumed that size- $l_{d}$ droplets are formed by liquid extending a distance $y=l_{d}$ inward from the jet boundary. Combined with the substitution of $l_{e}$ for $l_{d}$ and $t$ for $t_{s}$ based on prior assumptions, the expression for $t_{s}$ becomes $t\sim l_{e}/u_{\unicode[STIX]{x1D70F}}$ . This allows $l_{e}$ to be evaluated using (2.1), giving $l_{e}\sim \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{2}$ , which is substituted back into $t\sim l_{e}/u_{\unicode[STIX]{x1D70F}}$ to obtain $t\sim \unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}u_{\unicode[STIX]{x1D70F}}^{3}$ . Then, (2.3) is used to obtain

Substituting this into (2.2) and using the definition of $We$ ,

is obtained. Likewise, (2.3), the scaling of $l_{e}$ , the definition of $We$ and substitution of $l_{d}$ for $l_{e}$ give

In § 3, it is shown that the $We$ dependences predicted by (2.5) and (2.6) are well supported by measurements, but the measurements give no indication of $Re$ dependences as strong as predicted. In fact, they do not exclude the possibility that there is no dependence on  $Re$ .

Formally, the absence of $Re$ dependence can be obtained by assuming $V=U$ instead of $V=u_{\unicode[STIX]{x1D70F}}$ in the relation $t\sim L/V$ , which eliminates the $Re$ dependences without affecting the $We$ dependences. In this regard, note that here and also in the HIT analysis, the dependence of $L$ on $l_{e}$ (and a power of $D$ that maintains dimensional consistency) determines, through (2.1), the $We$ dependence of $l_{e}/D$ and consequently of $x_{i}/D$ and $l_{d}/D$ , while the velocity scale $V$ that is adopted determines the $Re$ dependences of $x_{i}/D$ and $l_{d}/D$ .

This raises the question of what physical mechanism could explain the dependence $t\sim y/U$ . A definitive answer is not yet in hand, but a plausible candidate is the near-field structure of the mean pressure field. In the fully developed nozzle flow that has been implicitly assumed, the mean pressure gradient is constant and axially oriented. Upon exiting the nozzle, the flow becomes transient, and so the mean pressure field also varies radially. Since the flow is subsonic, mean pressure effects extend over large scales, implying possible large-scale adjustments of flow features such as the velocity normalization in the law of the wall. The available large-scale velocity is the bulk velocity $U$ . This motivates the hypothesis that such adjustment suppresses the inlet-flow $Re$ dependence partially or entirely while preserving the logarithmic cross-stream structure near the jet perimeter. This hypothesis is testable through either measurements or detailed numerical simulations, which are therefore recommended as further steps towards complete understanding of near-field turbulent jet behaviours that affect the onset of breakup.

3 Comparison with measurements

In the published experimental studies whose implications are revisited here, care was taken to assure that the nozzle flow closely approximated fully developed pipe flow, as explained, e.g., by WF and Wu, Miranda & Faeth (Reference Wu, Miranda and Faeth1995). This does not guarantee that the flow state downstream of the nozzle orifice is fully determined by the nozzle flow. The injector tube necessarily has a finite thickness, implying the possible formation of one or more recirculation zones downstream of the end face of the tube. For the high liquid-to-gas density ratios in the experiments, the high inertia of the jet implies that these are gas-phase recirculation zones. This is supported by the image in figure 1, which suggests negligible near-field influence of the ambient air on the jet. Farther downstream, any aerodynamic coupling should be further mitigated by jet-induced gas flow, which tends to reduce interfacial shear that might affect the liquid flow. However, these attributes can be sensitive to details such as the tube thickness and the degree of planarity and smoothness of the tube face. The possibility that some liquid is entrained into the recirculation zones, which would increase their momentum coupling to the jet, cannot be ruled out. These points illustrate that it is difficult to construct a precisely canonical experimental configuration. In this regard, detailed numerical simulations of cases with nominally zero-thickness tubes might be advantageous for obtaining definitive resolution of the ambiguities noted in § 2.

Figure 1.

Image of near-field breakup in a 95 % ethanol jet issuing from a 1 cm diameter nozzle at a bulk velocity of $15~\text{m}~\text{s}^{-1}$ , corresponding to $Re=72\,000$ and $We=78\,000$ .

WF presented the results of several experimental studies in a plot of $x_{i}/D$ versus $We$ . The reported measurements of breakup-onset location are replotted in figure 2, along with line segments representing the HIT prediction $We^{-2/5}$ and the BL prediction $We^{-1}$ . The BL prediction is in better agreement with the measurements than the HIT prediction.

Figure 2.

Measured Weber-number dependence of the normalized breakup-onset location $x_{i}/D$ from the data compilation by WF. Data points designated by open squares are deemed to be outliers, as explained in the text. Solid line, $We^{-1}$ dependence (BL prediction); dashed line, $We^{-2/5}$ dependence (HIT prediction).

WF do not state the $Re$ values corresponding to the data points, but they can be inferred from their plotted data, information in the plot legends and the definitions of $We$ and $Re$ . On this basis, figure 3 is obtained. The vertical axis is compensated so that any apparent dependence on $Re$ cannot be caused by a correlation between the $We$ and $Re$ values of the chosen experimental cases. The plot gives no definitive indication of $Re$ dependence, though it is possible that a mild dependence is masked by the statistical scatter of the data.

Figure 3.

Reynolds-number dependence of the $We$ -compensated normalized breakup-onset location. The $Re$ values of the individual data points are deduced from information provided by WF, as explained in the text. Data points designated by open squares correspond to the similarly designated data points in figure 2. Solid line, $Re^{3/8}$ dependence (BL prediction).

To reduce this scatter, selected outlier cases relative to the $We^{-1}$ correlation line, designated by open squares in figure 2, are similarly designated in figure 3. The clustering of the outliers in both figures suggests that the cause of their deviations from the overall trends in these figures is systematic rather than statistical. Because these points are farthest from the $We^{-1}$ correlation line in figure 2, the compensation factor used in figure 3 is least effective in removing their $We$ dependence. This contributes to their deviations from collapse of the data in figure 3 as a function of $Re$ . Without these cases, it is seen that a systematic dependence on $Re$ can be ruled out with increased precision. On this basis, the $Re^{3/8}$ dependence predicted by the BL analysis, shown in the plot, is clearly inconsistent with the measurements, although the data can be plausibly interpreted as indicating a weaker dependence. In what follows, the theoretical interpretation is based on the default assumption that $x_{i}/D$ does not depend on $Re$ .

Figure 4.

Measurements of the Weber-number dependence of the droplet SMD at onset, normalized by the cross-stream integral scale $\unicode[STIX]{x1D6EC}$ , from the data compilation by SF. Solid line, $We^{-1}$ dependence (BL prediction); dashed line, $We^{-3/5}$ dependence (HIT prediction).

SF report measurements from various sources as well as their own measurements of droplet SMD at onset as a function of $We$ . They normalized the SMD using the cross-stream integral scale $\unicode[STIX]{x1D6EC}$ , for which they adopted the circular-pipe empirical correlation $\unicode[STIX]{x1D6EC}=D/8$ . Their data are replotted in figure 4 with superimposed line segments whose slopes correspond to the HIT and BL predictions.

SF propose that the droplet diameter should correlate with the ligament diameter, for which they obtain $We^{-0.94}$ dependence based on a best fit to data. They then show that this empirical dependence likewise fits their droplet SMD data very well. (This procedure is used rather than a direct fit to the droplet data because the ligament data conform to power-law behaviour more closely than the droplet data.) The $We^{-0.94}$ dependence that they thus infer is substantially closer to the BL prediction than to their HIT prediction.

Insufficient information is provided to back-calculate $Re$ for the individual points, so $Re$ dependence cannot be evaluated quantitatively. However, the fairly good collapse of the data as a function of $We$ suggests that any $Re$ sensitivity is likely to be weak, which does not necessarily exclude the $Re^{1/4}$ dependence predicted by the BL analysis.

The correlation lines in figures 2 and 4 serve not only to demonstrate that the BL predictions are more consistent with measured $We$ dependences than the HIT predictions, but also to provide improved theory-based inputs to empirical models of jet breakup that might be used in engineering and other applications. For this purpose, the correlations

and

corresponding to the solid lines plotted in the respective figures, are recommended.

An additional perspective on the BL analysis is obtained by expressing the typical droplet size at onset in wall units. The BL analysis assumes that the distance from the jet perimeter at which shear acts to form a droplet scales with the droplet size, and that this location is within the log layer of the precursor nozzle flow. For $Re=10^{4}$ , the location at which the log layer is deemed to begin is nominally 30 in wall units (Pope Reference Pope2000). The log law shows little $Re$ dependence at $Re$ values higher than this, so this lower bound on the logarithmic region is a generic high- $Re$ value.

Based on $\unicode[STIX]{x1D6EC}=D/8$ and the data in figure 4, 0.01 is taken to be a representative value of $l_{d}/D$ . The Blasius friction law can be expressed to a good approximation as $u_{\unicode[STIX]{x1D70F}}=0.2URe^{-1/8}$ . It follows that $l_{+}\equiv l_{d}u_{\unicode[STIX]{x1D70F}}/\unicode[STIX]{x1D708}=0.2(l_{d}/D)Re^{7/8}$ . This exceeds 30 for $Re>6\times 10^{5}$ , which is more that four times larger than the high end of the $Re$ range of the SF experimental cases. This brings into question the applicability of the BL analysis to the measurements.

There are several ambiguities that obscure the interpretation of this result. One is that the liquid region that is subject to the droplet-forming shear might not be detached in its entirety to form a droplet. Droplet release might involve an action–reaction mechanism in which a portion of the liquid rebounds and thus remains with the jet. Another is that large-scale transients possibly associated with the absence of $Re$ dependence of the breakup-onset location might also negate the applicability of standard law-of-the-wall phenomenology to the estimation of the extent of the log layer. These points further motivate future study of the details of breakup onset in order to clarify the governing mechanisms.