2 Flow set-up and parameters

The two-dimensional flow initially consists of two Gaussian vortices $i=1,2$ , separated by a distance $b_{0}$ (figure 1). Each vortex has an initial circulation $\unicode[STIX]{x1D6E4}_{i,0}=\unicode[STIX]{x03C0}a_{i,0}^{2}\unicode[STIX]{x1D714}_{i,0}$ , where $a_{i,0}$ is the initial vortex radius and $\unicode[STIX]{x1D714}_{i,0}$ is the initial peak vorticity, giving an initial aspect ratio $a_{2,0}/b_{0}$ and circulation ratio $\unicode[STIX]{x1D6EC}_{0}=\unicode[STIX]{x1D6E4}_{1,0}/\unicode[STIX]{x1D6E4}_{2,0}\leqslant 1$ . The simulations cover three subcategories of asymmetric pairs: those having unequal vorticity peaks but equal areas (UPEA: $\unicode[STIX]{x1D6EC}_{0}=\unicode[STIX]{x1D714}_{1,0}/\unicode[STIX]{x1D714}_{2,0}<1$ ), those having equal peaks but unequal areas (EPUA: $\unicode[STIX]{x1D6EC}_{0}=a_{1,0}^{2}/a_{2,0}^{2}<1$ ) and those having both unequal peaks and unequal areas (UPUA: $\unicode[STIX]{x1D6EC}_{0}=a_{1,0}^{2}\unicode[STIX]{x1D714}_{1,0}/a_{2,0}^{2}\unicode[STIX]{x1D714}_{2,0}<1$ ), as well as the limiting case of a symmetric pair ( $\unicode[STIX]{x1D6EC}_{0}=a_{1,0}^{2}/a_{2,0}^{2}=\unicode[STIX]{x1D714}_{1,0}/\unicode[STIX]{x1D714}_{2,0}=1.0$ ). The circulation Reynolds number, $Re_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}_{2,0}/\unicode[STIX]{x1D708}=5000$ , is based on the stronger vortex. Additional UPEA simulations are performed with $Re_{\unicode[STIX]{x1D6E4}}=1000$ in order to compare with the experimental data. All simulations are performed with $a_{2,0}/b_{0}=0.157$ . This value satisfies computational requirements (see Brandt & Nomura Reference Brandt and Nomura2007) and is comparable to those in the experiments. The simulations performed are summarized in tables 1 and 2. All times are given in terms of a convective time scale based on the period of revolution of two point vortices $T=2\unicode[STIX]{x03C0}^{2}b_{0}^{2}/(0.5\unicode[STIX]{x1D6E4}_{2,0}(1+\unicode[STIX]{x1D6EC}_{0}))$ , $t^{\ast }=t/T$ .

Figure 1.

The initial condition consisting of two Gaussian vortices $i=1,2$ each having initial radius $a_{i,0}$ , peak vorticity $\unicode[STIX]{x1D714}_{i,0}$ , separated by peak–peak distance $b_{0}$ . Initial aspect ratio $a_{2,0}/b_{0}=0.157$ and circulation ratio $\unicode[STIX]{x1D6EC}_{0}=\unicode[STIX]{x1D6E4}_{1,0}/\unicode[STIX]{x1D6E4}_{2,0}\leqslant 1$ .

Table 1.

Summary of simulation parameters and results for $Re_{\unicode[STIX]{x1D6E4}}=5000$ . Values for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ are evaluated at $t_{end}^{\ast }$ . Starting $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}$ is evaluated at $t_{start}^{\ast }$ . See § 4 for details and definitions of $t_{start}^{\ast }$ and $t_{end}^{\ast }$ . $\ast$ indicates that the straining-out regime has been reached and $\unicode[STIX]{x1D702}$ is therefore not a meaningful physical quantity. ‘Weak winner’ indicates that the peak of the weaker initial vortex becomes the peak of the resulting post-interaction vortex (see § 5).

The flow is incompressible and is integrated in time using finite-difference schemes on a uniform grid with periodic boundary conditions (see Gerz, Schumann & Elghobashi Reference Gerz, Schumann and Elghobashi1989; Orlandi Reference Orlandi2000). The domain consists of $2048^{2}$ points with side lengths of $L_{x}=L_{z}=24b_{0}$ , giving approximately 27 points across the larger core. Resolution and domain size testing on both local and integrated quantities showed these values to be sufficient; for further discussion see Brandt & Nomura (Reference Brandt and Nomura2007).

3 Basic flow behaviour

The flow development of three illustrative cases, a symmetric and two UPEA asymmetric pairs ( $\unicode[STIX]{x1D6EC}_{0}=1.0,0.90,0.70$ ), is presented in figure 2, with vorticity contour plots given in each of the upper rows of figure 2(a–c). Although only UPEA asymmetric pairs are shown, these cases depict the key processes involved in all interactions, which were investigated in detail by Brandt & Nomura (Reference Brandt and Nomura2007, Reference Brandt and Nomura2010). Here, we review these processes and consider further details in order to facilitate the current analysis.

Figure 2.

Vorticity contours at the initial condition and key times during the interaction for (a) symmetric ( $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6EC}_{0}=1.0$ ) and UPEA (b) $\unicode[STIX]{x1D6EC}=0.88$ ( $\unicode[STIX]{x1D6EC}_{0}=0.90$ ) and (c) $\unicode[STIX]{x1D6EC}=0.68$ ( $\unicode[STIX]{x1D6EC}_{0}=0.70$ ) cases at $Re_{\unicode[STIX]{x1D6E4}}=5000$ . For each case, the upper row depicts the vorticity contours of the total flow and the lower row depicts the vorticity contours within the region considered ‘core’ using $II>II_{t}$ . Column 1: initial condition; column 2: $t^{\ast }=t_{start}^{\ast }$ ; column 3: $t^{\ast }=t_{det}^{\ast }$ ; column 4: $t^{\ast }=t_{ent}^{\ast }$ (omitted for $\unicode[STIX]{x1D6EC}=0.68$ since no entrainment process occurs); column 5: axisymmetrization. See § 4 for details and definitions of $t_{start}^{\ast }$ , $t_{det}^{\ast }$ and  $t_{ent}^{\ast }$ .

Figure 3.

Time development of the normalized translational speed $|V|/|V_{0}|$ of each of the vorticity peaks, based on the initial translational speed of each vortex $V_{0}$ , for the cases depicted in figure 2. (a) $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6EC}_{0}=1.0$ , (b) $\unicode[STIX]{x1D6EC}=0.88\;(\unicode[STIX]{x1D6EC}_{0}=0.90)$ , (c) $\unicode[STIX]{x1D6EC}=0.68\;(\unicode[STIX]{x1D6EC}_{0}=0.70)$ .

Figure 4.

Trajectories of vortex peaks for the cases depicted in figure 2. Red $\times$ : vortex 1 (weaker), blue ○: vortex 2 (stronger). Initial positions indicated by filled square and circle respectively. Plotted data points correspond to intervals of approximately 200 time steps.

The mutually induced flow of the two vortices initially causes the vortices to revolve about each other with a nearly constant rotation rate ( ${\approx}2\unicode[STIX]{x03C0}/T$ ) and separation. These features are indicated in figure 3, which shows the translational speed of the vorticity peak of each vortex, and figure 4, which shows the trajectories of the vorticity peaks, for each of the cases shown in figure 2. The flow begins in a diffusive phase in which the flow evolves from the initial condition (column 1 of figure 2) primarily by viscous spreading of each vortex, which increases core size. The vortices also deform elliptically in response to the induced strain field of the other vortex. In the co-rotating frame, in the vicinity of the hyperbolic points where the vorticity magnitude is low, a tilt in the vorticity contours with respect to the streamlines develops (Brandt & Nomura Reference Brandt and Nomura2006). This results in filamentation at the outer ends of the vortex pair. At the central (hyperbolic) point, the tilt results in detrainment of core vorticity which is circulated and exchanged between the vortices. It is the establishment of this core detrainment process that is the predominant cause of the change from diffusion-dominated to convective-dominated behaviour, and is considered to be the beginning of the primary convective interaction.

In the symmetric pair (figure 2 a), the detrainment process is reciprocal and both vortices are detrained at the same rate (figure 2 a, columns 2–3). This weakens the vortex cores and at some point, they become mutually entrained (figure 2 a, columns 3–4) which corresponds to a rapid movement of the cores towards each other (figure 3 a). The original vortices are destroyed as they intertwine (figure 4 a) and the flow is transformed into a single compound structure (figure 2 a, column 4), which continues to axisymmetrize (figure 2 a, column 5).

In the asymmetric pairs (figure 2 b,c), the greater induced strain rate at the weaker vortex results in earlier deformation and detrainment (figure 2 b,c, columns 2–3). As the disparity of the vortices increases, more significant detrainment and even destruction of the weaker vortex may occur before detrainment can be established at the stronger vortex. The implication is that either a mutual but unequal entrainment process occurs ( $\unicode[STIX]{x1D6EC}_{0}=0.90$ , figure 2 b, columns 3–4) or no entrainment occurs ( $\unicode[STIX]{x1D6EC}_{0}=0.70$ , figure 2 c, column 3). In the former case, there is some intertwinement (figure 4 b) with only the weaker vortex destroyed. The stronger vortex entrains vorticity from the weaker vortex thereby yielding a compound vortex. In the latter case, there is no apparent interpenetration of vorticity from the original vortices (figure 4 c); the weaker vortex is destroyed while the stronger vortex remains, surrounded by vorticity from the weaker vortex.

The entrainment mechanism can be understood by considering the influence that each vortex exerts on the other. Huang (Reference Huang2005) elucidated this in a Lagrangian study of symmetric merger which showed that the vorticity that is detrained from one core and advected around the companion core, induces motion of the companion core towards it. In the symmetric case, since the cores are equal, the process is reciprocal and results in the rapid motion of the vorticity peaks towards each other as they are mutually entrained, as observed in figures 3(a) and 4(a) ( $\unicode[STIX]{x1D6EC}_{0}=1.0$ ). In the asymmetric case, as discussed in Brandt & Nomura (Reference Brandt and Nomura2010), the outcome depends on the relative time scales of each core’s detrainment and destruction. Consider two vortices 1 and 2 with $\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}<1$ . If vortex 1 is moderately weaker than vortex 2, although vortex 2 will detrain vortex 1 more quickly, vortex 1 will still detrain vortex 2, and the associated detrained vorticity will advect it inwards towards vortex 2. Since both vortices detrain, subsequently a mutual but non-reciprocal entrainment process will occur. This is observed in figures 3(b) and 4(b) ( $\unicode[STIX]{x1D6EC}_{0}=0.9$ ) which show vortex 1 accelerating with its trajectory moving inward while vortex 2 maintains its induced motion. Since vortex 1 is eventually eroded, its advection inward is limited, and only fluid from vortex 2 will become the centre of the resulting flow (figure 4 b). If vortex 1 is significantly weaker than vortex 2, it will be unable to detrain vortex 2 before it is eroded away by vortex 2 and thus will not be advected inwards. Hence, in this case, entrainment will not occur. This is seen in figures 3(c) and 4(c) ( $\unicode[STIX]{x1D6EC}_{0}=0.7$ ), wherein the peak of vortex 1 accelerates somewhat as it weakens but undergoes no inward motion, while vortex 2 decelerates as it establishes itself as the sole vortex in the flow. This is consistent with the results presented in a Lagrangian study of a highly asymmetric pair with $\unicode[STIX]{x1D6EC}_{0}=\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}=0.5$ by Huang (Reference Huang2006).

In all cases, whether or not entrainment occurs, by the end of the primary convective phase there exists only a single vortex structure, comprised of fluid from one or both starting cores, shown in column 4 of figure 2 for $\unicode[STIX]{x1D6EC}_{0}=1.0$ , 0.9 cases and in column 3 for $\unicode[STIX]{x1D6EC}_{0}=0.7$ case (since no entrainment occurs). Beyond the convective phase, the resulting structure undergoes a final phase (column 5 of figure 2) during which it axisymmetrizes into a circular Gaussian vortex which continues to evolve by viscous diffusion.

4 Assessing the product of asymmetric interactions

In order to quantitatively assess the interaction, an $\unicode[STIX]{x1D716}$ -type enhancement factor is evaluated,

and, since $\unicode[STIX]{x1D700}$ ( $\unicode[STIX]{x1D6E4}_{end}$ ) depends directly on the total available circulation of the starting pair, a corresponding merging efficiency,

is also considered. Both quantities are based on the circulation of the vortex, $\unicode[STIX]{x1D6E4}_{end}$ , at the end time of the interaction ( $t_{end}^{\ast }$ ) relative to the circulation of the stronger vortex, $\unicode[STIX]{x1D6E4}_{2,start}$ , and of the total circulation of both vortices combined, $\unicode[STIX]{x1D6E4}_{tot,start}$ , respectively, at the starting time of the interaction ( $t_{start}^{\ast }$ ). Since the key processes that lead to a single vortex occur in the primary convective phase, the vortex pair interaction is considered to be the primary convective interaction only (see appendix A), and the start and end times are considered to be those of this phase.

In order to evaluate (4.1) and (4.2) for viscous flow, the vortex cores must be identified in a consistent manner throughout the entire flow development including the transition from two vortices into one. To do this, a criterion is used based on the second invariant of the velocity gradient tensor: $II=1/2(\unicode[STIX]{x1D714}^{2}/2-S^{2})>II_{t}$ , where $II_{t}$ is an arbitrary threshold level (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). For the present study, the threshold level is set by $II_{t}^{\ast }\equiv II_{t}(t)/II_{peak}(t)=0.10$ , where $II_{peak}(t)$ is the instantaneous value of $II$ at the location of the higher of the values of $II_{i}$ evaluated at the location of each vorticity peak $i=1,2$ (or at the location of the only peak, if only one exists) unless otherwise noted. The value $II_{t}^{\ast }=0.10$ corresponds to the characteristic radius of a single isolated Gaussian vortex based upon the second moment of vorticity, $r=a_{\unicode[STIX]{x1D714}}$ , and is therefore used to identify the cores throughout the present study (Saffman Reference Saffman2001; Meunier et al. Reference Meunier, Ehrenstein, Leweke and Rossi2002). For more discussion of the choice of $II_{t}$ , see appendix A. Each lower row of figure 2 depicts the portion of the flow considered to be core by $II>II_{t}$ . It is seen that low-level peripheral and filamentary vorticity is excluded and the transformation of the flow from two distinct structures into one is captured.

Since the interaction between two isolated co-rotating vortices in viscous fluid always ultimately results in a single vortex, aggregate area and circulation of the vortex core(s) are evaluated by

and

where $\text{d}A$ refers to an area element of fluid (properties of each individual vortex $i=1,2$ are computed similarly using a separate $II_{t,i}$ for each, not shown). While this means that the definition of the weaker vortex core is somewhat inconsistent across $\unicode[STIX]{x1D6EC}$ , by monitoring the properties continuously, the times of start ( $t_{start}^{\ast }$ ) and end ( $t_{end}^{\ast }$ ) of the interaction can be identified without presupposing when the vortices are destroyed or when a new one is formed. Illustrative plots of $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ are presented in figure 5 which correspond to the cases depicted in figure 2. The aggregate properties are normalized by their values at the initial condition. Equivalent plots for a single Gaussian vortex having the same $Re_{\unicode[STIX]{x1D6E4}}$ as one member of the symmetric pair ( $Re_{\unicode[STIX]{x1D6E4}}=5000$ ) are also included in figure 5(d) for reference. In this case, $A_{II}(t)$ increases linearly in time while $\unicode[STIX]{x1D6E4}_{II}(t)$ remains very nearly constant, as expected, which verifies the evaluation of (4.3) and (4.4). Prior to $t_{start}^{\ast }$ , each individual vortex’s $\unicode[STIX]{x1D6E4}_{II,i}$ and $A_{II,i}$ evolve similarly to those of the single vortex (not shown).

Figure 5.

Time development of normalized aggregate core area $A_{II}(t)/A_{II,0}$ and circulation $\unicode[STIX]{x1D6E4}_{II}(t)/\unicode[STIX]{x1D6E4}_{II,0}$ (see (4.3) and (4.4)) for (a–c) the cases depicted in figure 2 and (d) a single vortex with same $Re_{\unicode[STIX]{x1D6E4}}$ . The dots in (a–c) correspond to the times of the columns in figure 2. (a) Symmetric ( $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6EC}_{0}=1.0$ ), (b) $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}=0.88\;(\unicode[STIX]{x1D6EC}_{0}=0.90)$ , (c)  $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}=0.68\;(\unicode[STIX]{x1D6EC}_{0}=0.70)$ , (d) single vortex.

In each of the pair interactions in figure 5(a–c), at first $A_{II}(t)$ grows linearly by viscous diffusion (i.e. during the first diffusive phase), while $\unicode[STIX]{x1D6E4}_{II}(t)$ decreases slightly due to the intensifying induced strain. There is then a deviation from linear growth of $A_{II}(t)$ due to filamentation and core detrainment. Since the start of detrainment is considered to be the start of the main convective interaction, $t_{start}^{\ast }$ is chosen to be the earlier time at which the evolution of either of the individual $A_{II,i}(t)$ begins a continual deviation from linear growth, reaching and thereafter maintaining at least a 0.5 % deviation from the $A_{II,i}$ value predicted by continual linear growth (note that computed $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ are relatively insensitive to choice of $t_{start}^{\ast }$ since $\unicode[STIX]{x1D6E4}_{II}$ remains approximately constant prior to detrainment, see appendix A for details). This essentially corresponds to the departure from linear growth of aggregate $A_{II}(t)$ but is somewhat easier to detect in UPUA cases. Note that for the remainder of this study, unless otherwise noted, vortex parameters such as $\unicode[STIX]{x1D6E4}_{II,1}$ , $\unicode[STIX]{x1D6E4}_{II,2}$ , and $\unicode[STIX]{x1D6EC}$ refer to quantities evaluated at $t_{start}^{\ast }$ , and that cases are identified by these properties (e.g. $\unicode[STIX]{x1D6EC}=0.88$ rather than $\unicode[STIX]{x1D6EC}_{0}=0.90$ ).

Subsequent to $t_{start}^{\ast }$ in figure 5(a–c), $\unicode[STIX]{x1D6E4}_{II}(t)$ rapidly decreases to a local minimum, at time $t_{det}^{\ast }$ , as vorticity detrains from the cores (column 3 of figure 2 corresponds to this minimum). For vortices similar in strength (figure 5 a,b), immediately after the time the $\unicode[STIX]{x1D6E4}_{II}(t)$ local minimum is reached, both $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ grow rapidly, with $\unicode[STIX]{x1D6E4}_{II}(t)$ quickly reaching a sharp peak, at time $t_{ent}^{\ast }$ , before diminishing again (column 4 of figure 2 corresponds to this maximum). For vortices having more disparate strengths (figure 5 c), after $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ reach their (simultaneous) local minima they both experience smooth, nonlinear behaviour without any spike, and ultimately return to single Gaussian vortex behaviour which is maintained until the end of the simulation. Note that there is not a sharp or sudden change in qualitative behaviour: intermediate cases contain qualitative elements of both (not pictured; these cases are treated as spike-occurring cases in all subsequent analysis).

Comparing figures 5(a,b) and 3(a,b), it is seen that the spikes in the symmetric and $\unicode[STIX]{x1D6EC}_{0}=0.90$ cases coincide with the speed increase that is associated with mutual entrainment as described in § 3. It is also seen that there is no spike or rapid acceleration in the UPEA $\unicode[STIX]{x1D6EC}_{0}=0.70$ case for which no entrainment occurs. The spike in the $\unicode[STIX]{x1D6E4}_{II}(t)$ plot is therefore associated with entrainment, in these cases $t_{end}^{\ast }$ is chosen to be $t_{ent}^{\ast }$ since at this time there exists only one continuous $II>II_{t}$ region, i.e. effectively one vortical structure (see lower rows of figure 2 a,b). Then for cases having no spike, $t_{end}^{\ast }$ is chosen to be $t_{det}^{\ast }$ since at that point there is already only one continuous region because no entrainment occurs (see lower row of figure 2 c). If multiple local minima or maxima exist, $t_{det}^{\ast }$ and $t_{ent}^{\ast }$ are taken at the extremum with the smallest or largest $\unicode[STIX]{x1D6E4}_{II}$ , as appropriate.

With $t_{start}^{\ast }$ and $t_{end}^{\ast }$ determined, $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ can be computed and results are discussed in the following sections. Alternative choices of $t_{start}^{\ast }$ and $t_{end}^{\ast }$ were considered and found to yield similar results in terms of both values and trends of $\unicode[STIX]{x1D700}$ (and $\unicode[STIX]{x1D702}$ ) (see appendix A for details). It is noted that $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ continue to evolve subsequent to $t_{end}^{\ast }$ , and careful observation shows that this corresponds to the axisymmetrization of the combined (i.e. final) structure and the diffusion of this structure into the non-quiescent post-interaction surrounding fluid (see appendix A). Since this predominantly viscous process occurs after the conclusion of the convective phase, it is not considered to be a part of the main interaction of the two vortices (i.e. not a part of the convective merging) and is therefore not considered in the evaluation of the interaction.

6 Relative straining and the mutuality parameter

The relationship between the metrics characterizing the interaction outcome ( $\unicode[STIX]{x1D700}$ and  $\unicode[STIX]{x1D702}$ ) and the basic parameters of the starting vortex pair (i.e. $\unicode[STIX]{x1D6EC}$ , $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}$ , $A_{1}/A_{2}$ ) presented in figure 7 and table 1 is unclear. A more effective parametrization may be developed by considering the underlying physics of the interaction. The analysis is given below and results are presented in figures 9–12. These figures include the $Re_{\unicode[STIX]{x1D6E4}}=5000$ cases as well as the $Re_{\unicode[STIX]{x1D6E4}}=1000$ cases. This section will focus on the $Re_{\unicode[STIX]{x1D6E4}}=5000$ results except where indicated; discussion of the $Re_{\unicode[STIX]{x1D6E4}}=1000$ results and viscous effects will be given in § 7.

Figure 9.

Results for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ versus mutuality parameter, $MP$ : (a) $\unicode[STIX]{x1D700}$ versus $MP$ and (b) $\unicode[STIX]{x1D702}$ versus $MP$ , for all cases. For $Re_{\unicode[STIX]{x1D6E4}}=5000$ cases, ▿: UPEA; ▵: EPUA; ♢: UPUA, stronger wins; $\times$ : UPUA, weaker wins; ○: no winner. For the $Re_{\unicode[STIX]{x1D6E4}}=1000$ cases, ▪: UPEA; ●: no winner.

As discussed in § 3, for vortex merger to occur, i.e. for a compound vortex with $\unicode[STIX]{x1D700}>1$ to be produced, mutual entrainment must occur to some extent so that there is some interpenetration of vorticity of the two vortices. This requires detrainment to be established in both vortices. If the disparity between the vortices is too great, detrainment will only be initiated in one vortex, which will then erode and be destroyed before it can induce detrainment of the other vortex, leaving it unenhanced, i.e. $\unicode[STIX]{x1D700}\sim 1$ . Since detrainment of a vortex is initiated by the induced strain and resulting deformation of vorticity, it may be characterized by a ratio of the strain rate to vorticity, $(S/\unicode[STIX]{x1D714})_{i}$ , which is here referred to as the relative straining. Here, $S_{i}$ and $\unicode[STIX]{x1D714}_{i}$ are evaluated at the location of the vorticity peak of vortex $i$ at the start of convective interaction $t_{start}^{\ast }$ (note $S=\sqrt{\boldsymbol{S}^{2}}$ where $\boldsymbol{S}^{2}=\text{tr}[S_{mn}S_{mn}]$ ). A mutuality parameter $MP$ is then defined as the ratio of the relative straining of each vortex,

which indicates the degree of mutuality of the interaction. A value of $MP=1$ corresponds to a fully two-way interaction. When $MP\sim 1$ , the vortices experience comparable relative straining so that both vortices are detrained, resulting in entrainment and $\unicode[STIX]{x1D700}>1$ . As $MP$ deviates from unity, it is expected that the interaction will become increasingly dominated by one vortex and $\unicode[STIX]{x1D700}$ will decrease until some critical value is reached, beyond which $\unicode[STIX]{x1D700}\sim 1$ , i.e. a one-way interaction.

Figure 9 shows $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ as functions of $MP$ . Note that figure 9 includes all the cases in figure 7 (i.e. $Re_{\unicode[STIX]{x1D6E4}}=5000$ ) as well as $Re_{\unicode[STIX]{x1D6E4}}=1000$ cases. For the $Re_{\unicode[STIX]{x1D6E4}}=5000$ results, the previous scatter in both $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ across all cases (UPEA, EPUA and UPUA) is seen to be significantly reduced, indicating a more effective characterization than $\unicode[STIX]{x1D6EC}$ . The dependence is peaked at $MP=1$ and generally monotonic to either side with both $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ decreasing as $MP$ varies from 1.

The majority of results correspond to $MP>1$ ; in this case, since $(S/\unicode[STIX]{x1D714})_{1}>(S/\unicode[STIX]{x1D714})_{2}$ , the weaker vortex detrains first and the stronger vortex then dominates. The associated values of $\unicode[STIX]{x1D700}$ decrease as $MP$ increases from 1, at first relatively gradually and then quite rapidly near a critical value of $MP$ , beyond which they remain constant at $\unicode[STIX]{x1D700}\sim 1$ . The critical value, $MP_{cr}\approx 1.78\pm 0.04$ , thus gives a criterion for merger in these cases. These interactions may then be characterized as follows: for $MP=1$ , symmetric merger (and in principle asymmetric merger with $(S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}=1$ , see figure 10) occurs and $\unicode[STIX]{x1D700}$ exhibits peak values; for $1<MP<MP_{cr}$ , $\unicode[STIX]{x1D700}>1$ , asymmetric merger occurs resulting in the stronger vortex being enhanced; and for $MP>MP_{cr}$ , $\unicode[STIX]{x1D700}\sim 1$ , straining out occurs with the stronger vortex surviving unenhanced. The behaviour of $\unicode[STIX]{x1D702}$ is similar, although for $MP<MP_{cr}$ , the decline is somewhat more muted; asymmetric merger is seen to be relatively high efficiency (most having $\unicode[STIX]{x1D702}>0.85$ ) indicating relatively minor losses for mergers in general.

When $MP<1$ , $(S/\unicode[STIX]{x1D714})_{1}<(S/\unicode[STIX]{x1D714})_{2}$ , and the corresponding cases in figure 9 are asymmetric mergers in which the weaker vortex dominates (as seen in figure 8 b) and is ultimately enhanced by the interaction, i.e. wins (computed $\unicode[STIX]{x1D700}$ is based on the stronger starting vortex regardless of which wins). It is expected that straining out will occur for $MP<MP_{cr}\approx 1/1.78=0.56$ , however, resolution considerations preclude further investigation of this weak-winner regime. When $MP\approx 1$ , neither vortex dominates or clearly wins (identified as ‘no winner’ in figure 9). Recall that in the symmetric case where $MP=1$ , both vortices are destroyed as they become mutually entrained and the flow transforms into a new vortex. This scenario can also occur in asymmetric (UPUA) cases when the relative straining on each vortex is nearly equal. An example of a no-winner UPUA case is given in figure 10, which shows that neither vortex dominates and the interaction proceeds in a similar manner to the symmetric case (figure 2 a). The symmetric and no-winner UPUA $MP\approx 1$ cases in the present study are seen to produce similar $\unicode[STIX]{x1D702}$ , implying comparably effective merging.

Figure 10.

Vorticity contours at the initial condition and key times during the interaction for a UPUA case ( $\unicode[STIX]{x1D6EC}=0.85$ , $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}=1.10$ , $Re_{\unicode[STIX]{x1D6E4}}=5000$ ) in which no winner can be determined ( $MP=1.02$ ). See figure 2 for meanings of the rows and columns.

It is useful to relate $MP$ to the previously considered vortex parameters. Simple expressions for $(S/\unicode[STIX]{x1D714})_{i}$ can be obtained by considering the external strain rate produced by a point vortex, i.e. the strain rate induced by one vortex at the location of the other were the other not present, e.g. $S_{1}=\unicode[STIX]{x1D6E4}_{2}/(2\unicode[STIX]{x03C0}b^{2})$ (a similar expression can be obtained using the expression for strain rate produced by a Lamb–Oseen vortex Saffman Reference Saffman2001), and using $\unicode[STIX]{x1D6E4}_{i}=\unicode[STIX]{x03C0}a_{i}^{2}\unicode[STIX]{x1D714}_{i}$ ,

This relates $(S/\unicode[STIX]{x1D714})_{i}$ to the vortex aspect ratios. For symmetric and UPEA pairs, equations (6.2)–(6.3) imply $MP=1/\unicode[STIX]{x1D6EC}^{2}$ , which is observed to hold approximately true across the range of $\unicode[STIX]{x1D6EC}$ considered in the present study (not shown). The critical $MP_{cr}\approx 1.78$ then implies a critical $\unicode[STIX]{x1D6EC}_{cr}\approx \sqrt{1/1.78}=0.75$ , very close to the observed UPEA $\unicode[STIX]{x1D6EC}_{cr}\approx 0.79$ (figure 7).

A simple relation for $MP$ may then be obtained from (6.2)–(6.3),

where $\unicode[STIX]{x1D714}_{i}^{2}a_{i}^{2}$ is the enstrophy of a Rankine vortex. This suggests that the enstrophy ratio at the onset of convective interaction may capture the same physical principles and consequent $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ behaviour as $(S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}$ . More generally, the enstrophy ratio can be defined as

where the integral is evaluated for each vortex at $t_{start}^{\ast }$ . Figure 11 shows $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ plotted against $Z_{2}/Z_{1}$ ; the observed dependence is similar to that of $MP$ . It is noted, however, that this parameter is somewhat less successful at indicating the weak-winner regime boundary (figure 11). The critical enstrophy ratio beyond which no enhancement occurs is found to be $(Z_{2}/Z_{1})_{cr}\approx 1.63\pm 0.03$ for $Re_{\unicode[STIX]{x1D6E4}}=5000$ .

Figure 11.

Results for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ versus enstrophy ratio, $Z_{2}/Z_{1}$ : (a) $\unicode[STIX]{x1D700}$ versus $Z_{2}/Z_{1}$ and (b) $\unicode[STIX]{x1D702}$ versus $Z_{2}/Z_{1}$ , for all cases. For the $Re_{\unicode[STIX]{x1D6E4}}=5000$ cases, ▿: UPEA; ▵: EPUA; ♢: UPUA, stronger wins; $\times$ : UPUA, weaker wins; ○: no winner. For the $Re_{\unicode[STIX]{x1D6E4}}=1000$ cases, ▪: UPEA; ●: no winner.

To the authors’ knowledge, the only other study to consider strong-winner and weak-winner interactions is Melander et al. (Reference Melander, Zabusky and McWilliams1987b ), who found a transition region between these cases in their parameter space, which they approximated by $(\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2})=(A_{1}/A_{2})^{-0.36}$ . A similar expression can be obtained from (6.4) with $MP=1$ (the boundary for the weak-winner regime in the present study), $(\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2})=(A_{1}/A_{2})^{-0.5}$ , which approximately corresponds to the ‘lower edge’ of the transition region of Melander et al. (see their figure 2). Considering the slightly different definitions of ‘winner’ and the presence of viscosity in the present study, this correspondence is notable.

Examining $(S/\unicode[STIX]{x1D714})_{i}$ of each vortex separately, it is observed that the convective interaction begins when the relative straining of the first vortex to detrain reaches some consistent value. Figures 12(a) and 12(b) show the starting $(S/\unicode[STIX]{x1D714})_{1}$ and the maximum value of $(S/\unicode[STIX]{x1D714})_{2}$ reached before the end of detrainment, respectively, plotted against the corresponding $MP$ for all numerical cases. For each case the interaction outcome is coded as binary ‘merger’ or ‘straining out’ based on whether an entrainment process is evident in the $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ plots (this classification is consistent with a $\unicode[STIX]{x1D700}$ -based assessment, but avoids invoking an arbitrary $\unicode[STIX]{x1D700}$ cutoff to classify the $\unicode[STIX]{x1D700}\approx 1$ cases). Considering the results at both $Re_{\unicode[STIX]{x1D6E4}}=5000$ and $Re_{\unicode[STIX]{x1D6E4}}=1000$ , it is observed that entrainment only occurs if the stronger vortex achieves a maximum $(S/\unicode[STIX]{x1D714})_{2,max}\geqslant 0.138$ prior to the end of detrainment and never occurs for $(S/\unicode[STIX]{x1D714})_{2,max}\leqslant 0.132$ , suggesting a critical $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ (see figure 12 b), which is comparable to the relatively consistent mean value of starting $(S/\unicode[STIX]{x1D714})_{1}=0.147$ with an average deviation of 0.016 (figure 12 a). The scatter in the starting $(S/\unicode[STIX]{x1D714})_{1}$ values in the present study is attributed to imprecision in determining the exact starting time (see § 4). Thus, the critical value $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ generally characterizes the onset of detrainment of Gaussian vortices having $Re_{\unicode[STIX]{x1D6E4}}$ in the range considered in this study.

Figure 12.

Results for the relative straining at each vortex with indication of occurrence of detrainment: (a) $(S/\unicode[STIX]{x1D714})_{1}$ versus $MP$ at $t_{start}^{\ast }$ , (b) maximum $(S/\unicode[STIX]{x1D714})_{2}$ prior to the end of detrainment $t_{det}^{\ast }$ versus $MP$ . $Re_{\unicode[STIX]{x1D6E4}}=5000$ : ▫: merger, $+$ : straining out. $Re_{\unicode[STIX]{x1D6E4}}=1000$ : ♢: merger, $\times$ : straining out. Dashed line indicates the critical $(S/\unicode[STIX]{x1D714})_{cr}=0.135,$ obtained from (b) (see § 6). Merger and straining out assessed by occurrence of entrainment phase and lack thereof respectively.

A critical strain rate to vorticity ratio has also been considered in a number of related studies. Le Dizès & Laporte (Reference Le Dizès and Laporte2002) and Le Dizès & Verga (Reference Le Dizès and Verga2002) showed that a similar quantity evaluated in the co-rotating frame is a leading-order approximation of the eccentricity of streamlines near the centre of an elliptical Gaussian vortex, i.e. a local eccentricity. Brandt & Nomura (Reference Brandt and Nomura2010) identified a critical value of their strain rate parameter (the strain rate evaluated at the central hyperbolic point in the co-rotating frame) that characterized the start of detrainment. Their results also show that start of detrainment corresponds to an approximately consistent local eccentricity $\unicode[STIX]{x1D700}_{l}\approx 0.2$ at the start of interaction (compare their figures 6a and 8), equivalent to $(S/\unicode[STIX]{x1D714})\approx 0.14$ (although here $(S/\unicode[STIX]{x1D714})$ is not evaluated in the co-rotating frame), which is similar to the observed $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ in the present study. Mariotti, Legras & Dritschel (Reference Mariotti, Legras and Dritschel1994) considered the similar phenomenon of vortex stripping of a single vortex by external adverse shear and likewise found a comparable critical ratio of strain rate to vorticity, equivalent to $S/\unicode[STIX]{x1D714}\approx 0.095$ in the present study, and moreover found this value to be consistent over a range of $Re_{\unicode[STIX]{x1D6E4}}$ , i.e. that this ‘stripping’ (and ultimately vortex breakup) is an inviscid mechanism, even in viscous flow. Trieling et al. (Reference Trieling, Velasco Fuentes and van Heijst2005) utilized a similar critical strain rate to characterize the boundary between complete and partial straining-out regimes in inviscid flow, relating the critical shear required to destroy the weaker vortex to a critical separation distance. This effectively distinguished partial (two vortices survive) and complete (one vortex completely destroyed) straining-out cases in inviscid flow for EPUA and UPEA pairs having small radius ratio $R_{2}/R_{1}$ and small peak vorticity ratio $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}$ , respectively, for distributed vortices having various vorticity distributions, including a Gaussian profile. Since viscous effects in the present study reduce $\unicode[STIX]{x1D714}$ in $(S/\unicode[STIX]{x1D714})$ , the critical value $(S/\unicode[STIX]{x1D714})_{cr}$ (directly related to critical separation distance for symmetric merger; see (6.2)–(6.3)) is always attained by at least one of the vortices which is ultimately destroyed, precluding two-vortex outcomes. Trieling et al. (Reference Trieling, Velasco Fuentes and van Heijst2005) attributed the inefficacy of this criterion for pairs of low asymmetry (i.e. those having higher $R_{2}/R_{1}$ or $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}$ ) to spatial effects and captured circulation by the stronger vortex altering its induced strain. In light of the present analysis, it is suggested that this could be attributed to the mutual detrainment of both vortices activating the convective entrainment mechanism before either is completely destroyed (i.e. before the critical $(S/\unicode[STIX]{x1D714})$ for vortex destruction in inviscid flow is reached).

7 Comparison with experimental results and $Re_{\unicode[STIX]{x1D6E4}}$ effects

A series of UPEA numerical simulations at $Re_{\unicode[STIX]{x1D6E4}}=1000$ are performed in order to facilitate comparison with available experimental data. The simulation parameters and results are summarized in table 2. Contour plots for illustrative cases (symmetric $\unicode[STIX]{x1D6EC}=1.0$ and UPEA $\unicode[STIX]{x1D6EC}=0.88$ , 0.67), corresponding to those for $Re_{\unicode[STIX]{x1D6E4}}=5000$ in figure 2, are presented in figure 13. It is clear that diffusion and the onset of detrainment occur more rapidly for each case than in the comparable higher- $Re_{\unicode[STIX]{x1D6E4}}$ case. These basic effects are also seen in the evolutions of $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ (figure 14 a,b; symmetric case not shown). Otherwise, these results exhibit qualitatively similar features indicating that the flows experience the same underlying processes.

Table 2.

Summary of parameters and results for $Re_{\unicode[STIX]{x1D6E4}}=1000$ computational simulations (‘Num.’) and laboratory experiments (‘Expt.’). Here all numerical cases are UPEA cases. $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ are evaluated at $t_{end}^{\ast }$ . Starting $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}$ is evaluated at $t_{start}^{\ast }$ . Due to the nature of the experiments, only $\unicode[STIX]{x1D6EC}_{0}$ was controlled in the initial condition. $\ast$ indicates that the ‘straining-out’ regime has been reached and $\unicode[STIX]{x1D702}$ is therefore not a meaningful physical quantity. ‘Weak winner’ indicates that the peak of the weaker initial vortex becomes the peak of the resulting post-interaction vortex (see § 5). Margins of error for the experimental cases correspond to extrema of error bars in figure 16 (for details see § 7).

Figure 13.

Vorticity contours at the initial condition and key times during the interaction for (a) Symmetric ( $\unicode[STIX]{x1D6EC}_{0}=1.0$ ) and UPEA (b) $\unicode[STIX]{x1D6EC}=0.88$ ( $\unicode[STIX]{x1D6EC}_{0}=0.90$ ) and (c) $\unicode[STIX]{x1D6EC}=0.67$ ( $\unicode[STIX]{x1D6EC}_{0}=0.70$ ) cases at $Re_{\unicode[STIX]{x1D6E4}}=1000$ . The first row depicts the vorticity contours of the total flow, and the second row depicts the vorticity contours within the region considered ‘core’ using $II>II_{t}$ (see § 4). See figure 2 for meanings of columns.

Figure 14.

Time development of normalized aggregate core area $A_{II}(t)/A_{II,0}$ and circulation $\unicode[STIX]{x1D6E4}_{II}(t)/\unicode[STIX]{x1D6E4}_{II,0}$ ( $II>II_{t}$ ) for $Re_{\unicode[STIX]{x1D6E4}}=1000$ numerical cases (a) $\unicode[STIX]{x1D6EC}=0.88$ and (b) $\unicode[STIX]{x1D6EC}=0.67$ , and comparable experimental cases (c) $\unicode[STIX]{x1D6EC}\approx 0.97$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1500$ and (d) $\unicode[STIX]{x1D6EC}\approx 0.69$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1000$ . Note that the ‘initial’ time for (c) and (d), i.e. time of evaluation of $X_{II,0}$ quantities, corresponds to the time at which vortices are judged to have formed.

The most significant implication of decreasing $Re_{\unicode[STIX]{x1D6E4}}$ is to increase $MP_{cr}$ ( ${\approx}2.15\pm 0.16$ ) and increase $(Z_{2}/Z_{1})_{cr}$ ( ${\approx}2.00\pm 0.14$ ) relative to the $Re_{\unicode[STIX]{x1D6E4}}=5000$ case, as indicated in the plots of $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ versus $MP$ and $Z_{2}/Z_{1}$ for $Re_{\unicode[STIX]{x1D6E4}}=1000$ (figures 9 and 11). Careful examination of the $\unicode[STIX]{x1D6E4}_{II}(t)$ and vorticity contour plots for all the simulations verifies that an entrainment process occurs in some low- $Re_{\unicode[STIX]{x1D6E4}}$ cases where there is none at the corresponding high $Re_{\unicode[STIX]{x1D6E4}}$ case with similar $\unicode[STIX]{x1D6EC}$ and $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}$ . As discussed in the previous section, $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ is consistent for both $Re_{\unicode[STIX]{x1D6E4}}$ considered (see figure 12). The increase of $MP_{cr}$ with decreasing $Re_{\unicode[STIX]{x1D6E4}}$ is therefore attributed to the more rapid diffusion enabling comparatively stronger vortices to achieve $(S/\unicode[STIX]{x1D714})_{cr}$ by reducing $\unicode[STIX]{x1D714}$ more quickly. This is consistent with the results of Mariotti et al. (Reference Mariotti, Legras and Dritschel1994) for a single vortex, which indicate that the critical $S/\unicode[STIX]{x1D714}$ for vortex breakup by erosion, found to be independent of $Re_{\unicode[STIX]{x1D6E4}}$ , is reached more quickly for lower $Re_{\unicode[STIX]{x1D6E4}}$ . Although the case of interacting vortices is more complex, the processes underlying the onset of detrainment are similar (Kimura & Herring Reference Kimura and Herring2001; Brandt & Nomura Reference Brandt and Nomura2006). Additional effects of reducing $Re_{\unicode[STIX]{x1D6E4}}$ may include an increase in $\unicode[STIX]{x1D700}$ values as a result of the earlier onset of detrainment potentially reducing the amount of filamentation that can occur (although such a reduction is difficult to quantify), as well as increased diffusion of vorticity directly from one vortex into the other (although this effect is considered to be minor since the vortices are well-separated prior to convective merging, which then occurs on a much more rapid time scale than diffusion).

The numerical results are compared with digital particle image velocimetry (DPIV) data taken from water tank experiments of co-rotating symmetric and asymmetric vortex pairs (see Meunier Reference Meunier2001; Meunier et al. Reference Meunier, Ehrenstein, Leweke and Rossi2002). The vortices are generated impulsively by the rotation of sharp-edged flat plates about their vertical axes such that they behave as laminar, two-dimensional vortices without axial velocity in the observation volume, with $\unicode[STIX]{x1D6EC}_{0}$ adjusted by varying the relative angular velocity of the plates. The experimental cases are summarized in table 2. The $\unicode[STIX]{x1D6EC}_{0}$ value is evaluated at the time at which the vortices are judged to have become coherent and detached from the flaps by visual examination (which may differ slightly from Meunier et al. Reference Meunier, Ehrenstein, Leweke and Rossi2002). The starting $\unicode[STIX]{x1D6EC}$ value in each case is the average of the highest and lowest of several possible $\unicode[STIX]{x1D6EC}$ values computed using different plausible choices of starting time in order to minimize the effect of the variability in these values (since with noisy data slight differences in choice of start/end times can have a relatively pronounced effect).

Contours of vorticity comparable to those in figure 13 are presented in figure 15 for the $\unicode[STIX]{x1D6EC}\approx 0.97$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1500$ and $\unicode[STIX]{x1D6EC}\approx 0.69$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1000$ experimental cases. The oblong structures visible at the early times are the plates that produce the vortices and should be disregarded. Despite the noise, these appear qualitatively similar to those of the comparable numerical cases.

Figure 15.

PIV results for experimental cases having (a) $\unicode[STIX]{x1D6EC}\approx 0.97$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1500$ and (b) $\unicode[STIX]{x1D6EC}\approx 0.69$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1000$ . The first row depicts the vorticity contours of the entire flow and the second row depicts the vorticity contours of the portion considered ‘core’ using $II>II_{t}$ . See figure 2 for meanings of columns. Note that the ‘initial condition’ corresponds to the time at which vortices are judged to have formed.

Plots of $A_{II}(t)$ and $\unicode[STIX]{x1D6E4}_{II}(t)$ for the experimental results are presented in figure 14(c,d) along with the $Re_{\unicode[STIX]{x1D6E4}}=1000$ numerical cases. No smoothing or averaging has been applied to the experimental data, but despite the noise they qualitatively resemble those of the comparable numerical cases: the $A_{II}(t)$ of both experimental cases suggest a linear growth at first, then nonlinear behaviour before returning to linear viscous growth, while $\unicode[STIX]{x1D6E4}_{II}(t)$ of both appears to grow slightly at first (attributed to the method used to create the vortices) followed by a decline to a local minimum, then a spike in the $\unicode[STIX]{x1D6EC}\approx 0.97$ case but not in the $\unicode[STIX]{x1D6EC}\approx 0.69$ case, followed by relatively smooth nonlinear behaviour. These features are interpreted to correspond to the diffusive, convective and final phases of interaction that are observed in the numerical cases. The longer diffusive phase in the experimental $\unicode[STIX]{x1D6EC}\approx 0.69$ case (compared to the numerical $\unicode[STIX]{x1D6EC}=0.67$ , $Re_{\unicode[STIX]{x1D6E4}}=1000$ case) is attributed to its having a somewhat lower initial aspect ratio $a_{0}/b_{0}$ .

Results for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ for the experimental cases are summarized in table 2 and plotted with the low- and high- $Re_{\unicode[STIX]{x1D6E4}}$ numerical results in figure 16. As with $\unicode[STIX]{x1D6EC}$ , for each case several $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ were computed using a number of plausible choices for ending times due to the noise in the data. The integrated quantity $Z_{2}/Z_{1}$ is used to examine the trends since the local quantities $(S/\unicode[STIX]{x1D714})$ and $MP$ are too uncertain to give meaningful results. The experimental results in figure 16 are shown with horizontal error bars indicating the maximum and minimum plausible $Z_{2}/Z_{1}$ and vertical error bars indicating the maximum and minimum potential $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ within this range, and $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ plotted at the midpoint. It is noted that one experimental case ( $\unicode[STIX]{x1D6EC}\approx 0.92$ with $\unicode[STIX]{x1D714}_{1}/\unicode[STIX]{x1D714}_{2}>1$ , $Re_{\unicode[STIX]{x1D6E4}}\approx 1100$ ) is a UPUA weak-winner case. Despite the limitations of the data, the experimental values are in reasonable accordance with the numerical results.

Figure 16.

Results for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ versus enstrophy ratio, $Z_{2}/Z_{1}$ : (a) $\unicode[STIX]{x1D700}$ versus $Z_{2}/Z_{1}$ and (b) $\unicode[STIX]{x1D702}$ versus $Z_{2}/Z_{1}$ , for all cases. For the $Re_{\unicode[STIX]{x1D6E4}}=5000$ cases, ▿: UPEA; ▵: EPUA; ♢: UPUA, stronger wins; $\times$ : UPUA, weaker wins; ○: no winner. For the $Re_{\unicode[STIX]{x1D6E4}}=1000$ cases, ▪: UPEA; ●: no winner. The experimental results are indicated by ${\vartriangleleft}$ with error bars, see § 7 for explanation.

8 Summary and discussion

The interactions of two unequal co-rotating vortices in viscous fluid have been further investigated and for the first time, their outcomes quantitatively assessed. A new parameterization of the possible interaction outcomes has been developed. Numerical simulations were performed to consider a range of vortex strength ratios, $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}\leqslant 1$ , corresponding to vortices of differing size and/or peak vorticity, and with $Re_{\unicode[STIX]{x1D6E4}}=5000$ . An additional set of simulations at lower $Re_{\unicode[STIX]{x1D6E4}}$ was also performed in order to compare with available data from the experiments of Meunier (Reference Meunier2001).

The primary convective interaction between the two vortices is initiated by the induced strain of each vortex on the other, and in viscous flow, leads to a single vortex. The post-interaction vortex is quantitatively assessed in terms of an enhancement factor, $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D6E4}_{end}/\unicode[STIX]{x1D6E4}_{2,start}$ , and a corresponding merging efficiency, $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D6E4}_{end}/\unicode[STIX]{x1D6E4}_{total,start}$ , similar to the analysis of inviscid flow by Dritschel & Waugh (Reference Dritschel and Waugh1992), which compare the vortex strengths at the start and end of the convective interaction. An effective parameterization is developed by characterizing the degree of mutuality of the interaction with the mutuality parameter, $MP\equiv (S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}$ , where the relative straining $(S/\unicode[STIX]{x1D714})_{i}$ , evaluated at the start of the interaction, is an indicator for core detrainment of each vortex.

Analysis of the simulation results demonstrates the general description: two vortices will undergo mutual entrainment (merger) if $MP$ is sufficiently close to unity, i.e. within some critical value, $MP_{cr}$ . For the present results, $MP_{cr}\approx 1.8$ for $MP>1$ (stronger vortex dominates). Asymmetric merger with $MP_{cr}<MP<1$ may also occur, and in this case the originally weaker vortex dominates and survives enhanced. Symmetric merger ( $MP=1$ ) is the limiting case of a fully two-way reciprocal interaction with $\unicode[STIX]{x1D700}\sim 2$ and correspondingly, $\unicode[STIX]{x1D702}\sim 1$ ; the most efficient merger. As $MP$ increasingly deviates from unity, both $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ diminish until $MP\sim MP_{cr}$ , beyond which $\unicode[STIX]{x1D700}\approx 1$ and the interaction is essentially one-way; entrainment does not occur and the dominating vortex erodes the other away (straining out) and survives relatively unaffected. The enstrophy ratio $Z_{2}/Z_{1}$ is found to characterize the behaviour of $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ similarly to $MP$ overall, although it is somewhat less effective at distinguishing the weak-winner regime. Results also indicate that core detrainment is initiated when $(S/\unicode[STIX]{x1D714})_{i}$ reaches a critical value of $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ . A simple analysis shows that $(S/\unicode[STIX]{x1D714})_{i}$ can be related to $\unicode[STIX]{x1D6EC}$ and $a_{i}/b$ . In the case of symmetric merger, this is consistent with the initiation of detrainment corresponding to the critical aspect ratio, $(a/b)_{cr}$ , which is inevitably achieved in viscous flow. This may then also explain, more generally, why the outcome of interactions in viscous flow is always a single vortex: diffusion of core vorticity will weaken the cores so that eventually $(S/\unicode[STIX]{x1D714})_{cr}$ is attained by at least one of the vortices, which in time becomes completely eroded.

These observations generally hold for the lower $Re_{\unicode[STIX]{x1D6E4}}$ simulations and for the available experimental data. Since initiation of detrainment is essentially an inviscid mechanism, we may expect the value of $(S/\unicode[STIX]{x1D714})_{cr}$ to remain constant with $Re_{\unicode[STIX]{x1D6E4}}$ , as the present results support. However, the value of $MP_{cr}$ (and $(Z_{2}/Z_{1})_{cr}$ ) is found to vary somewhat (favouring merger) as $Re_{\unicode[STIX]{x1D6E4}}$ decreases. Since $MP$ may be considered a ratio of time scales, i.e. the disparity in timing for detrainment to be established in each of the vortices, this is influenced by viscous effects. With decreasing $Re_{\unicode[STIX]{x1D6E4}}$ , the more rapid diffusion of vorticity will accelerate the achievement of $(S/\unicode[STIX]{x1D714})_{cr}$ so that more disparate vortices may still both detrain within the interaction time.

The present results have implications for two-dimensional turbulence. The observation that in asymmetric interactions, one of the original vortices essentially survives with its peak becoming that of the post-interaction vortex, may explain the noted persistence of vorticity extrema throughout the evolution of turbulent flow (McWilliams Reference McWilliams1990; Carnevale et al. Reference Carnevale, McWilliams, Pomeau, Weiss and Young1991, Reference Carnevale, McWilliams, Pomeau, Weiss and Young1992; Dritschel et al. Reference Dritschel, Scott, Macaskill, Gottwald and Tran2008). Moreover, the findings of the present study show how the peak of the post-interaction vortex can be related directly to the original pair, which in principle allows for modelling and perhaps analytical consideration of two-dimensional turbulent flows having vortices of varying peak vorticity. Such analysis remains for future work.

The present findings also suggest that the simple transformation rule utilized in a number of point vortex models of two-dimensional turbulence to incorporate vortex merging may not accurately reflect the enhancement produced by interactions of real vortices. This rule is derived from conservation of energy and typically takes the form $\unicode[STIX]{x1D6E4}_{A}^{2}+\unicode[STIX]{x1D6E4}_{B}^{2}=\unicode[STIX]{x1D6E4}_{C}^{2}$ , where $A$ and $B$ are the interacting vortices and $C$ is the resulting vortex, which ‘replaces’ $A$ and $B$ to emulate merging (Carnevale et al. Reference Carnevale, McWilliams, Pomeau, Weiss and Young1991; Riccardi et al. Reference Riccardi, Piva and Benzi1995; Sire et al. Reference Sire, Chavanis and Sopik2011). This expression can be written in terms of enhancement as $1+\unicode[STIX]{x1D6EC}^{2}=\unicode[STIX]{x1D700}_{model}^{2}$ by dividing by the $\unicode[STIX]{x1D6E4}$ of the stronger initial vortex, and it can be seen that this generally underestimates the enhancement produced by symmetric or slightly asymmetric mergers (e.g. for symmetric pairs, $\unicode[STIX]{x1D700}_{model}=1.41$ but observed $\unicode[STIX]{x1D700}\approx 2$ ), but overestimates the enhancement produced by very asymmetric pairs (e.g. for the $\unicode[STIX]{x1D6EC}=0.7$ UPEA case, $\unicode[STIX]{x1D700}_{model}=1.22$ but observed $\unicode[STIX]{x1D700}\approx 1$ ). It is emphasized that this theory was never intended to be precise at the level of individual interactions, but it is nevertheless at odds with the findings of isolated vortex pair studies. Trieling et al. (Reference Trieling, Velasco Fuentes and van Heijst2005) reported a similar finding and proposed that conservation of circulation may be more accurate, although they found this overestimates the enhancement produced by straining-out interactions, a finding supported by the rapid diminishing of $\unicode[STIX]{x1D702}$ with increasing $MP$ (and $Z_{2}/Z_{1}$ ) observed here.

Based on the present findings, it is suggested that a simple binary transformation rule could improve accuracy without adding prohibitive complication: when $(S/\unicode[STIX]{x1D714})$ of a vortex in a two-dimensional turbulent flow exceeds $(S/\unicode[STIX]{x1D714})_{cr}$ ( ${\approx}0.135$ ), this value is compared with the $(S/\unicode[STIX]{x1D714})$ of the nearest vortex to determine $MP$ . If $MP<MP_{cr}$ , the merger conserves circulation, i.e. the two vortices $A$ and $B$ are replaced with a single vortex $C$ having circulation $\unicode[STIX]{x1D6E4}_{C}=\unicode[STIX]{x1D6E4}_{A}+\unicode[STIX]{x1D6E4}_{B}$ . If $MP>MP_{cr}$ , the stronger’s circulation is preserved, i.e. $\unicode[STIX]{x1D6E4}_{C}=\unicode[STIX]{x1D6E4}_{A}$ , where $\unicode[STIX]{x1D6E4}_{A}>\unicode[STIX]{x1D6E4}_{B}$ . Further study is needed to determine $MP_{cr}$ as a function of $Re_{\unicode[STIX]{x1D6E4}}$ , but it is expected that for most typical flows values of $MP_{cr}$ in the range presented here will constitute reasonably accurate choices. A more nuanced transformation rule, perhaps incorporating the continuous variation of $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D702}$ with $MP$ and/or incorporating the concept of interaction winners, could also be considered. This remains for future work.

One major aspect of two-dimensional turbulence not considered in this study is the influence of the surrounding vortices in the flow on the interacting pair, which could be simply approximated by an imposed external strain. Although a few studies have considered the effect of a background flow on the interactions of symmetric vortex pairs (see, e.g. Maze et al. Reference Maze, Carton and Lapeyre2004; Perrot & Carton Reference Perrot and Carton2010; Trieling, Dam & van Heijst Reference Trieling, Dam and van Heijst2010; Folz & Nomura Reference Folz and Nomura2014), none have considered the case of asymmetric pairs. Such a study, using the present analysis framework, is currently in progress by the authors.