2.6.1. Short impulse (regimes I and IV): linear growth

Comparing the two time scales involved in (2.25), i.e. the impulse duration $\tau _d$ versus the natural oscillation period $\tau _p=2{\rm \pi}$ associated with the GCW, we assume in the following that $\tau _d \ll \tau _p$. In physical units, the natural period obtained for the critical mode using (2.15) is independent of the velocity scale $U_m$, given by

(2.26)\begin{equation} T_0=\frac{\sqrt{2}{\rm \pi} \sigma^{1/4}(\rho_1+\rho_2)^{1/2}}{[g(\rho_1-\rho_2)]^{3/4}}.\end{equation}

Accordingly, the peak amplitude $\eta _{max}=\max _{\tau \geq 0}|\hat {\eta }(\tau )|=|\hat {\eta }(\tau _m)|$ occurs at a time $\tau _m>\tau _d$, and its magnitude corresponds to that of the waveform (2.25), read as

(2.27)\begin{equation} \eta_{max}^2=\frac{|\hat{\eta}_d|^2+{|\hat{\eta}'_d|^2}}{2}+ \sqrt{\frac{(|\hat{\eta}_d|^2-{|\hat{\eta}'_d|^2})^2}{4}+ [{{\rm Im}(\hat{\eta}_d){\rm Im}(\hat{\eta}'_d)+{\rm Re}(\hat{\eta}_d) {\rm Re}(\hat{\eta}'_d)}]^2}.\end{equation}

Further, for short impulses of $\tau _d \to 0$, the asymptotic behaviour of $\hat {\eta }_d$ and $\hat {\eta }'_d$ is captured well by the first-order truncated solution $\hat {\eta }_1$ given in (2.20), leading to

(2.28a,b)\begin{equation} \hat{\eta}_1(\tau_d)=1-i \tau_d-\frac{\tau_d^2}{2}+\alpha \tau_d M^2 S_\infty,\quad \hat{\eta}'_1(\tau_d)={-}i-\tau_d+M^2S_\infty,\end{equation}

where the assumption $S(\tau _d)=S_\infty$ is used, and for each given $U_2$ as a function of time, the constant fraction

(2.29)\begin{equation} \alpha=\frac{1}{\tau_d S_\infty}\int_0^{\tau_d} S(\tau)\, {\rm d} \tau\in (0, 1],\end{equation}

is introduced when $U_2\leq 1$ in (2.21). Substituting (2.28a,b) into (2.27) thus generates the following limit as $\tau _d\to 0$:

(2.30)\begin{equation} \eta_{max}\sim \sqrt{1+\frac{ M^4 S_\infty^2+M^2S_\infty\sqrt{M^4S_\infty^2+4}}{2}}\sim 1+\frac{M^2 S_\infty}{2}+o(S_\infty), \end{equation}

that is independent of $\alpha$, the impulse's profile action correction factor. Therefore, at leading order, the maximum interfacial perturbation caused by a given imposed airflow is completely determined by its total action $S_\infty$, regardless of its temporal profile. In the small $S_\infty \to 0$ limit, we will show, in § 2.7.2 that, for all four $U_2$ impulse profiles of (2.16), with formally different $\tau _d$, $S_\infty$ and $\alpha$ given in table 1, a common linear increase of $\eta _{max}$ with corresponding $S_\infty$ emerges.

2.6.2. Long impulse (regimes II and III): exponential growth

Having established that $S_\infty$ is the key quantity intrinsic to $U_2$ that governs the maximum interfacial growth for sort impulses, we investigate next the behaviour of $\eta _{max}$ for larger $\tau _d$ when the first-order approximation $\hat {\eta }_1(\tau _d)$ no longer holds. We derive asymptotic results for the step ($S$), linear ($L$) and exponential ($E$) profiles to demonstrate that $\eta _{max}$ grows exponentially with respect to $S_\infty$ in this regime at a rate that is dependent on the detailed $U_2$ shape. Note that the Gaussian ($G$) profile is excluded in this calculation due to difficulties in evaluating the associated (2.22) in closed form, but is confirmed numerically to be consistent with the other three profiles considered here (see § 2.7.2)

The step profile, $U_2^S$ given by (2.16a) in the limit of $\tau _1=0$, is considered first. We obtain the exact solutions to $\hat {\eta }$ and $\hat {\eta }'$ in this case by either integrating and summing (2.22), or solving (2.14) directly, leading to the exponential solution

(2.31) \begin{equation} \hat{\eta}^S(\tau)=\cosh \left(\tau \sqrt{{M}^2-1}\right)-\frac{i \sinh \left(\tau \sqrt{M^2-1}\right)}{\sqrt{M^2-1}}. \end{equation}

Therefore, substituting $\hat {\eta }_d=\hat {\eta }^S(\tau _2)=\hat {\eta }^S(S_\infty )$ into (2.27) gives the desired result

(2.32)\begin{equation} \eta^S_{max}\sim\frac{M^2 \exp\left({S_\infty\sqrt{M^2-1}}\right)}{2\sqrt{M^2-1}},\end{equation}

as $S_\infty \to \infty$, where it is also evident that $\eta ^S_{max}\propto M$ for large $M$.

Next, we examine the linear and the exponential profiles, given by (2.16b) and (2.16c), respectively, using $\tau _1=0$ and $M\to \infty$ (regime III). These limits are taken to enable analytic evaluation of (2.22). Following the derivations detailed in § A.3, one arrives at asymptotic expressions for $\hat {\eta }^{L,E}$ and ${\eta }_{max}^{L,E}$ analogous to (2.31) and (2.32). Particularly, again, in the limit of $M\to \infty$ (regime III), we show that

(2.33)\begin{equation} \eta^L_{max}\sim \frac{\displaystyle \varGamma\left(\frac{3}{4}\right)}{{(2{\rm \pi})}^{{1}/{2}} 3^{{1}/{4}}}\frac{M^{{3}/{4}}\exp\left({\dfrac{3 M S_\infty}{2}}\right)}{S_\infty^{{1}/{4}}},\end{equation}

for the linear profile $U_2^L$, and

(2.34)\begin{equation} \eta^E_{max}\sim \frac{M^{{1}/{2}} \exp(2MS_\infty)}{2\sqrt{\rm \pi} S_\infty^{1/2}}, \end{equation}

for the exponential profile $U_2^E$. In both cases, ${\eta }_{max}^{L,E}$ as a function of $S_\infty$ are again dominated by an exponential term.

The comparison between (2.32), (2.33) and (2.34) reveals that, although $\eta _{max}$ generated by the three different $U_2$ profiles grow similarly with an exponential pattern as $S_\infty$ increases, the exact asymptotic forms for the growth differ between the profiles for large $S_\infty$. Therefore, the total action of an imposed flow in this case, while remaining an important predictive variable, no longer uniquely determines the resulting interfacial perturbation. In this regime III, the dependence of $\eta _{max}$ on the impulse functional form details in this limit of $S_\infty \to \infty$, i.e. long impulses, is in contrast with that of regimes I and II in the limit of $S_\infty \to 0$ (§ 2.6.1), where we had found that $\eta _{max}$ is independent of the details of the impulse flow profile functional form (as we also confirm with numerical simulations in figure 6).

Figure 6.

(a) Profiles of imposed airflow $U_2(\tau )$ given in (2.16). Normalized time unit $T=\tau /\tau _p$ is used. A common flow peak time of $\tau _1/\tau _p=0.006$ and total action of $S_\infty =0.04$ are used with effective impulse strength $M=10$. This regime illustrates imposed short impulses, with $T_d\ll 1$, and large part of the impulse being classically KH unstable. The numerical solutions for the interfacial perturbation amplitude $|\hat {\eta }(\tau )|$ in response to the imposed airflow impulse shown in (b). Here, the interface response curves collapse on a single curve independent of the details of the impulse airflow functional form as predicted by the small action asymptotic solution we gave in (2.22), where $|\hat {\eta }|$ is uniquely determined by the total impulse action $S_\infty$ in the limit of $S_\infty \to 0$. This insight helps guide our thinking about the key quantity governing destabilization for transient impulses. However, the total action $S_\infty$ is not a universal quantity capturing the fate of the interface given an arbitrary impulse functional form as seen in panels (c) and (d), depending on the limits of $S_\infty$ and $M$ values, we note that amplification can start to depend on the details of the impulse functional form imposed, as seen in the insets.

2.7.1. Evolution of $\hat {\eta }$ for short and long impulses and importance of action $S_\infty$

Recall that from (2.16) we model the flow impulse $U_2$ using four different types of temporal profiles. Representative examples for each of the profiles are illustrated in figure 6(a). A normalized time unit $T\equiv \tau /\tau _p$, with respect to the interface's natural oscillation period $\tau _p=2{\rm \pi}$, or in physical units $\varPi$ given in (2.26), is used henceforth for all canonical impulses, so that the imposed flow duration $T_d = \tau _d/\tau _p$ and its effective strength $M$ (see (2.13)) can be independently determined by the dimensional maximum velocity and duration, respectively. Recall that the horizontal line positioned at $U_2=1/M$ divides the flow history into portions that are conventionally KH unstable ($MU_2>1$) and KH stable ($MU_2<1$). In figure 6, we discuss the response of the interface with respect to short impulses (regimes I and IV). In figure 7 we discuss the long ones.

Figure 7.

(a),(b) Profiles for the imposed airflow $U_2(\tau )$ given in (2.16) with long duration. Normalized time unit $T=\tau /\tau _p$ is used. A common flow peak time of $\tau _1/\tau _p=0.3$ and total action of $S_\infty =6.28$ are used in (a) and $\tau _1/\tau _p=0.4$, $S_\infty =8.38$ in (b). The effective impulse strength are $M=1.4$ and $M=0.9$, respectively in (a,b). (c,d) The numerical solutions for the interfacial perturbation amplitude $|\hat {\eta }(\tau )|$ in response to the imposed flow signals given in (a) and (b), respectively. Oscillatory interface response is established, where the maximum amplitude occurs at the first peak.

In figure 6(a), we show short impulses, defined with $T_d\ll 1$ and $1/M\ll 1$ and with a common impulse peak time $\tau _1/\tau _p=0.006$ as well as common total action $S_\infty =0.04$. We show the corresponding responses of the interface in figure 6(b). We see an overall sinusoidal behaviour of $|\hat {\eta }|$ for all canonical impulse airflows as expected from our derivation of (2.25). We also see a collapse of the interface response for different impulse airflow profiles. In other words, the response of the interface is independent of the details of the impulse's functional form and only depends on the total action $S_\infty$ as predicted by the small action asymptotic solution we derived in (2.22), where we showed $|\hat {\eta }|$ to be uniquely determined by $S_\infty$ in the limit of $S_\infty \to 0$ (extreme limits of regimes I and IV). This independence of the interface's response to the impulse airflow details remains true as long as $S_\infty \to 0$, regardless of $M$, as seen in figure 6(c,d). As $S_\infty$ increases, i.e. the impulse duration increases, figure 6(c,d) shows how the peak amplitude response $\eta _{max}$ starts differing between canonical impulse functional forms. Note, however, that, even then, the initial increase of $|\hat {\eta }|$ for small times during the impulse, e.g. $T\lesssim 0.1$, remains insensitive to the impulse's particular shape. Generally, this initial stage of the $|\hat {\eta }|$ growth is well captured by the series solution we derived in (2.22).

In figure 7(a,b), we show long impulses defined to have a duration comparable to the natural oscillation period: $T_d \gtrsim 1$ (regimes II and III). The responses to the impulses with strength $1/M=0.7$ and 1.1 are shown in figures 7(c) and 7(d), respectively. In figure 7(c), in contrast to the short impulse response (figure 6b), the impulse in figure 7(a) leads to an interface amplitude that peaks at its first oscillation and then decreases over time. The length of the impulse enables, for an initial amplification phase that lasts long enough, the creation of a higher first peak that ends up also being the global amplitude maximum $\eta _{max}$. This peak occurs at $T = T_m$ before the airflow impulse ends, i.e. $T_m< T_d$. This suggests that the decay of $U_2$ does not allow a larger interface perturbation beyond the first peak to be sustained. This transient growth and maximum initial peak amplification property is further discussed § 2.7.3. Note also that, in this regime, the $|\hat {\eta }|$ response of the interface is more sensitive to the details of the impulse functional form despite the use of a common total action, $S_\infty =6.28$. Thus, the total action $S_\infty$, as derived in § 2.6.1, is less of a universal predictor for $|\hat {\eta }|$ outside the asymptotic limit of large $M$ for long impulses ($S_\infty \to \infty$). Finally, the transient growth introduced in § 2.5 is shown in figure 7(b,d), where $M=0.9$. Hence, the entire flow impulse is considered KH stable (regime II), yet we see interface amplitude amplification in figure 7(d) for both impulses imposed. These are similar to the long impulse response of figure 7(c) with a maximum amplitude of the perturbation $\eta _{max}$ at a time $T_m< T_d$, followed by oscillations of decreasing amplitude. However, here, we also see more clearly the increase in frequency for subsequent oscillations.

2.7.2. Maximum amplitude $\eta _{max}$

The analytic theory for maximum perturbation amplitude $\eta _{max}$ developed in §§ 2.6.1 and 2.6.2 is considered next. By comparing $\eta _{max}$ generated by different $U_2$ profiles of common $S_\infty$, figure 6 shows that the numerical data clearly collapse for $S_\infty \to 0$. In this short impulse regime, we find an excellent agreement between the numerical results and the analytical estimate from (2.30), for both large and small flow strengths, $M$. This is illustrated in figure 6(c) with $M=50$ and figure 6(d) with $M=5$. The key insight is that the leading-order behaviour of $\eta _{max}$ associated with imposed flows of regimes I and IV with short $\tau _d$, or equivalently small $S_\infty$, is uniquely specified by $S_\infty$, independent of the details of the exact temporal profile of the impulse. However, such universality and reduction deteriorates for impulses of long duration, where we find increasing discrepancies among the $U_2$ profiles as $S_\infty$ increases (figure 6c,d).

In the large $S_\infty$ limit (regimes II and III), the $\eta _{max}$ approximations are shown in figure 8 for each flow impulse. Assuming $\tau _1\to 0$ for all four cases, the asymptotic $\eta _{max}$ takes the form

(2.35)\begin{equation} \eta_{max}\sim C\frac{M \exp({a M S_\infty})}{ (MS_\infty)^b},\end{equation}

where $C$, $a$ and $b$ are constants given by (2.32) and (2.33) and (2.34) for the step, linear and exponential profiles, respectively; and are fitted for the Gaussian profile as $C=0.36$, $a=1.35$, $b=0.11$, according to the numerical solutions obtained using $M=500$.

Figure 8.

Value of $\eta _{max}$ as a function of $S_\infty$, for a given $M$, obtained numerically (open circles) and its analytic approximations for both small and large $S_\infty$ limits, given by the dashed and solid lines, respectively (see (2.30), (2.35)). For each flow profile shown in (a)–(d), $\tau _1=0$ and three different values of $M$ are compared, showing a transition into the regime of exponential dependence of $\eta _{max}$ observed for large $S_\infty$.

Formally derived for impulses of high strength, i.e. $M\to \infty$ (regime III), the exponential growth described by (2.35) performs reasonably well matching the numerical solutions in figure 8, across all flows for a range of $M$ down to order unity (regimes II and III). These results suggest that, despite the differences in $C$, $a$ and $b$, the empirical form (2.35) holds true for the $U_2$ profiles considered. Together with the small $S_\infty$ theory given in (2.30), this establishes a full asymptotic description of $\eta _{max}$ for $M>1$.

Finally, note that our asymptotic and numerical results of exponential $\hat {\eta }$ growth so far relied on the assumption that the imposed impulse airflow profile reaches maximum velocity close to instantaneously at $t=0^+$, that is, $\tau _1=0$ and $U_2(0^+)=1$. We tested the sensitivity of our results to this assumption. This is shown in § A.1 with figure 21 showing that the maximum amplification, $\eta _{max}$, is essentially insensitive to the time ratio $\tau _1/\tau _d$ for a given impulse action $S_{\infty }$ and given intensity $M$ where amplification occurs. Figure 21 also supports that, for long impulses, of increasing $S_{\infty }$ values, $\eta _{max}$ is maximized when $\tau _1=0$. Physiologically, this observation implies that the $\tau _1/\tau _d\to 0$ limit of violent exhalation would be favoured to promote maximal interface perturbation. This is consistent with exhalation impulse flow profiles measured (Bourouiba Reference Bourouiba2021b) to be compatible with $\tau _1/\tau _d\ll 1$.

2.7.3. Amplitude peak time $T_m$

Next, we discuss the critical time $T_m$ at which the maximum interface amplitude first occurs. For flow impulses of short duration (regimes I and IV), we show using (2.25) and (2.28a,b) that $T_m\to 1/8$ as $T_d\to 0^+$, regardless of the $U_2$ temporal profile and the impulse strength $M$. For the regime I and III partly unstable flows illustrated in figure 9(a), we take the linear and Gaussian flow profiles in figures 9(b) and 9(c), respectively, to show that increasing the impulse duration $T_d$ (from regime I to III) increases $T_m$ which starts becoming greater than $T_d$, but is eventually exceeded by $T_d$ as $T_d$ increases. This means that the maximum interface amplitude, $\eta _{max}$, is reached after the impulse vanishes if the impulse is short (regime I), and before the flow impulse ends if the impulse is long (regime III). The exact transition time between these two regimes depends on $M$ and the flow profile details. Assuming $M>1$ and $T_d$ large, $T_m$ is also bounded below by

(2.36)\begin{equation} T_*=\max\left\{T=\frac{\tau}{2{\rm \pi}}: U_2(\tau)=M^{{-}1}\right\},\end{equation}

where $T_*$ is the transition time during the imposed airflow from being classically KH stable to unstable, giving $T_*< T_m< T_d$ as compared in figures 9(b) and 9(c) for both profiles. This is because $|\hat {\eta }|$ must exponentially increase during the period over which $MU_2>1$, based on (2.14). Further, the increase of $T_m$ with $T_d$ appears to be linear and unbounded for large $T_d$ when $M>1$, since $T_*\to \infty$ as $T_d\to \infty$. Note that these findings appear to be robust for both the linear and Gaussian profiles and especially insensitive to the cutoff value $d$ in (2.17) used to define $T_d$ in the latter case. We also obtain similar results for the exponential flow profile (not shown).

Figure 9.

Maximum amplitude time $T_m$ as a function of impulse duration $T_d$. The linear flow impulse profile is used in (a) and (d) and the Gaussian profile in (c) and (f), where $T_d$ is defined using $d=0.01$ in (2.17). For both profiles, $\tau _1=0$ is applied. The flow impulses imposed in (b) and (c) have effective strength $M>1$, while $M<1$ in (e) and (f). This is schematically illustrated in (a) and (b), respectively, where $T_d$ and $T_*$ (defined in (2.36)) are labelled along the velocity profile. In all plots, the solid lines of different colour give $T_*$ obtained with varying $M$. In (b) and (c), the upper bound $T_d$ is given by the dashed line, and the lower bound $T_*$ is shown as the dotted line. In (f), the large $T_d$ limit ((2.19)) is given by the dot-dashed lines. The figure shows that, for strong impulses ($M>1$), the interface peak amplification time occurs after the impulse relaxes into the classically KH stable regime, i.e. $T_m>T_*$. And for weak but long impulses ($M<1$, $T_d>O(1)$), a peri-impulse peak occurs ($T_m< T_d$) at a time bounded below by the limit $1/(4\sqrt {1-M^2})$ as $T_d\to \infty$ (§ 2.5).

In contrast, the amplitude peak time associated with transient growth where $M<1$ (part of regime II) does not increase indefinitely, as shown in figures 9(d)–9(f). Instead, because the flow is never classically KH unstable, $T_m$ must converge to the limiting value $T_m=1/(4\sqrt {1-M^2})$, and is independent of $T_d$, as we derived in (2.19) in the limits of $\tau _1\to 0$ and $T_d\to \infty$. Figure 9(f) shows the match between the numerical results and our asymptotic results.

In both cases ($M>1$ and $M<1$), the observation that $T_m< T_d$ for flow impulses with $T_d\gtrsim 1$ suggests again that, once the interface has reached its local maximum amplification during its first period, the following period is unable to generate a higher subsequent perturbation, rendering the first amplitude peak a global maximum. On the other hand, the lower bound $T_m>T_*$ implies that it is beneficial to have the impulse's peak time $\tau _1\to 0$ so that $T_*$ is minimized in order to induce maximum interface perturbation as early as possible.

3.1.1. The unsteady vortex method

First, we parameterize the periodic interface in complex plane using Lagrangian markers $a\in [0,1]$, such that the interface shape at time $t$ is given by $z=z(a,t)=x(a,t)+iy(a,t)$. We also introduce a vortex strength $\gamma (a,t)$ along the interface, a local jump in tangential velocity, ${\rm \Delta} u$, between the two fluids from below ($j=1$) to above ($j=2$), and cumulative circulation $\varGamma$ measured from $a=1$

(3.1a,b)\begin{equation} {\rm \Delta} u(a,t)= \frac{\gamma(a,t)}{s_a},\quad \varGamma(a,t)={-}\int_a^{1}\gamma(a',t) \, {\rm d} a'={-}\int_a^{1}{\rm \Delta} u(a',t)s_a(a',t)\, {\rm d} a',\end{equation}

where $s_a=\sqrt {x_a^2+y_a^2}$ is the arclength metric, with subscript $a$ denoting partial derivatives with respect to $a$. Periodicity in the $x$-direction then reads $z(a+2{\rm \pi},t)=z(a,t)+2{\rm \pi}$ and $\gamma (a+2{\rm \pi},t)=\gamma (a,t)$. This vortex sheet formulation of the interface, as shown in figure 10, enables us to construct the complex potential $W$, valid in both fluid regions

(3.2)\begin{equation} W(Z,t)=\phi+\psi i=\frac{1}{2{\rm \pi} i}\int_{0}^{1}\log\{\sin[{\rm \pi}(Z-z(a',t))]\}\gamma(a',t)\,{\rm d} a'+\bar{U}(t)Z, \end{equation}

where $\phi$ and $\psi$ are the velocity potential and streamfunction, respectively, $Z=X+iY$ represents any field point $(X,Y)$ in the complex plane and $\bar {U}=[U_1(t)+U_2(t)]/2$ is the average unsteady flow impulse between the two fluids (Pullin Reference Pullin1982). We note that, while $\psi$ is continuous everywhere, $\phi$ jumps across the interface, leading to

(3.3) \begin{equation} W_- - W_+=\phi_- - \phi_+ = \varGamma(a,t),\end{equation}

where the subscripts $\pm$ denote the $Z\to z$ limit taken from above and below the interface, respectively.

Figure 10.

Schematic of the periodic vortex sheet parameterization of a two-fluid interface. Here, $0\leq a\leq 1$ is the particle-like marker, $z(a,t)$ is the complex interface shape and $\gamma (a,t)$ gives the vortex strength distribution.

It also follows that the complex velocity field $q$ is given by the Birkhoff–Rott integral

(3.4)\begin{equation} q^*(Z,t)= \frac{\partial W}{\partial Z}= u-iv=\frac{1}{2i}\int_{0}^{1}\cot[{\rm \pi}(Z-z')]\gamma'\,{\rm d} a'+\bar{U}(t), \end{equation}

where again the asterisk denotes complex conjugate, and the shorthand notation $z'\equiv z(a',t)$, $\gamma '\equiv \gamma (a',t)$ are used. As such, the potential flow equation introduced in (2.1a) and the interface kinematic condition (2.3a) are automatically satisfied. The far-field limits of $q$ as $Y\to \pm \infty$ must match the imposed flow impulse, leading to the total circulation

(3.5)\begin{equation} \varGamma(0,t)=U_2(t)-U_1(t)=\frac{2A}{1+A}U_2(t),\end{equation}

where we used (3.1a,b) and (2.5). Further, we define the interface velocity specified by each marker $a$ as the weighted average of the complex velocities on either side of the interface, $q_{\pm }$, as follows:

(3.6)\begin{equation} \dot{z}^*(a,t)=\bar{q}^*+\frac{f\gamma}{2z_a}=\left\{\frac{1}{2 i}\int\kern-8pt -_{0}^{1}\cot[{\rm \pi}(z-z')]\gamma'\,{\rm d} a'+\bar{U}(t)\right\}+\frac{f\gamma}{2z_a},\end{equation}

where the dot symbol denotes partial derivative with respect to $t$ while fixing $a$, $\bar {q}^*=\bar {q}^*(a,t)$ is the mean flow velocity between $q^*_{\pm }$, following the Cauchy principal value integral, and $f\in [-1,1]$ is an arbitrary weight parameter. Taking $f=\pm 1$ recovers the discontinuous limiting flow velocities as $q_{\mp }^*=\bar {q}^*\pm \gamma /(2z_a)$.

For closure, we derive the evolution equation for $\gamma$ next. Evaluating the Bernoulli equation (2.1b) in the limit of $Z\to z_{\pm }$ and substituting into the dynamic condition (2.3b) generates

(3.7) \begin{align} -\frac{\kappa}{We}&=\frac{\partial(\phi_+ -\phi_{-})}{\partial t} -A \frac{\partial(\phi_{-}+\phi_+)}{\partial t}+ \frac{(1-A)|q_+|^2-(1+A)|q_-|^2}{2}-\frac{2A y}{{Fr}^2}\nonumber\\ &\quad +F_1(t)-F_2(t), \end{align}

where $\partial /\partial t$ is understood as the Eulerian derivative at constant $Z$. The interface curvature here can be expressed as

(3.8)\begin{equation} \kappa(a,t)=\frac{{\rm Im}\{(\partial z^*/\partial a)(\partial^2 z/\partial a^2)\}}{|\partial z/\partial a|^3}. \end{equation}

Substituting (3.2), (3.3) and the Lagrangian velocity (3.6) into (3.7) yields

(3.9)\begin{equation} \dot{\varGamma}={-}2A\dot{\bar{\phi}}+fA\gamma {\rm Re}\left\{\frac{\bar{q}}{z_a}\right\}+A|\bar{q}|^2+ \left(\frac{f}{2}-\frac{A}{4}\right)\frac{\gamma^2}{|z_a|^2} -\frac{2Ay}{{Fr}^2}+\frac{\kappa}{We}+F_1(t)-F_2(t),\end{equation}

where $\dot {\bar {\phi }}=(\dot {\phi }_+ + \dot {\phi }_-)/2$. Next, using the identity

(3.10)\begin{equation} \bar{\phi}_a={\rm Re}\{\bar{q}^* z_a\},\end{equation}

as well as (3.6), we can differentiate (3.9) with respect to $a$. After some algebra, we arrive at the following Fredholm equation of the second kind for $\dot {\gamma }$:

(3.11)\begin{equation} \dot{\gamma}+A\int\kern-8pt -_0^1{\rm Im}\{\dot{\gamma}'z_a\cot {\rm \pi}(z-z')\}\, {\rm d} a'=Q(a,t),\end{equation}

where $\dot {\gamma }'\equiv \dot {\gamma }(a,a')$ and

(3.12)\begin{equation} Q=\left(\frac{f}{2}-\frac{A}{4}\right)\frac{\partial}{\partial a} \left(\frac{\gamma^2}{|z_a|^2}\right)+A\left({\rm Im}\{z_a I\} +f\gamma{\rm Re}\left\{\frac{\bar{q}_a}{z_a}\right\} -2x_a\dot{\bar{U}}-\frac{2y_a}{{Fr}^2}\right)+\frac{\kappa_a}{We},\end{equation}

with

(3.13)\begin{equation} I=I(a,t)=\int\kern-8pt -_0^1 \frac{\gamma'{\rm \pi}(\dot{z}-\dot{z}')}{\sin^2 {\rm \pi}(z-z')}\,{\rm d} a',\quad \dot{z}'\equiv \dot{z}(a,a'). \end{equation}

Equations (3.6) and (3.11) form a system of IDEs that determines the evolution of the two-fluid interface. An alternative derivation for (3.11) (equivalent in principle but differing in formulations) can also be found in Pozrikidis (Reference Pozrikidis2011). Compared with the results of Pullin (Reference Pullin1982) and Baker et al. (Reference Baker, Meiron and Orszag1982), unsteadiness in the flow field here enters explicitly in terms of the average velocity $\bar {U}$ and acceleration $\dot {\bar {U}}$, so the flow impulse needs to be continuous and piecewise differentiable in time. This also differs from the linear theory (2.14) where $U_2$ is only required to be square integrable.

3.1.2. The vortex blob regularization

Although the present boundary integral techniques are popularly used to model vortex sheets, it is also well known that standard numerical discretizations can produce unstable schemes. This is partly due to the curvature singularity in the exact solution without surface tension. Therefore regularization of the Birkhoff–Rott integral in (3.6) is commonly used to overcome both the physical and numerical instabilities (Baker & Pham Reference Baker and Pham2006; Sohn et al. Reference Sohn, Yoon and Hwang2010).

Here, we implement the vortex blob method developed by Baker & Beale (Reference Baker and Beale2004), which allows control of physical instabilities in the fluid motion and permits calculation past the time of singularity formation by introducing a smoothing length scale $\delta$. This provides numerical damping that adds to surface tension that also acts as a dispersive regularization of the KH instability (Hou et al. Reference Hou, Lowengrub and Shelley1997; Baker & Nachbin Reference Baker and Nachbin1998). However, the dispersive, small-scale capillary waves associated with surface tension are possibly dampened as a result.

Specifically, we introduce the vortex blob by replacing the original periodic kernel $K=\cot {\rm \pi}(z-z')$ in (3.6) with the following modification:

(3.14)\begin{equation} K_\delta(a,a') = K(a,a')\left[1+g\left(\frac{r}{\delta}\right)\right], \end{equation}

where $\delta \ll 1$ is the smoothing length, and

(3.15a,b)\begin{equation} r=r(a,a')=\frac{|\sin{\rm \pi}(z-z')|}{\rm \pi}, \quad g(r/\delta)={-}\exp(-(r/\delta)^2). \end{equation}

We chose a Gaussian smoothing function, $g$, in order to ensure that, as $\delta \to 0$, the regularized vortex sheet approaches a classical weak solution of the Euler equations (Baker & Pham Reference Baker and Pham2006). Moreover, the simple poles in (3.6) and (3.11) can be removed using the following identity (Baker & Beale Reference Baker and Beale2004):

(3.16)\begin{equation} {\rm Im} \int\kern-8pt -_0^1 K_\delta(a,a')z_a'=\int\kern-8pt -_0^1 B(a,a')[1+g(r/\delta)]\, {\rm d} a'=0, \end{equation}

where $z_a'\equiv z_a(a,a')$ and

(3.17)\begin{equation} B=B(a,a')=\frac{-\{x_a'(2{\rm \pi}(x-x'))+y_a' \sinh[2{\rm \pi}(y-y')]\}}{2{\rm \pi} r^2}. \end{equation}

As a result, the regularized IDE system becomes

(3.18a)\begin{gather} \dot{z}^*=\frac{1}{2 i}\int_{0}^{1}\left\{\gamma'\cot{\rm \pi}(z-z') +\frac{\gamma B}{{\rm \pi} z_a}\right\}[1+g(r/\delta)]\,{\rm d} a' +\bar{U}(t)+\frac{f\gamma}{2z_a}, \end{gather}

(3.18b)\begin{gather}\dot{\gamma}+A \int_0^1{\rm Im}\left\{\dot{\gamma}'z_a\cot{\rm \pi}(z-z') +\frac{\dot{\gamma}B}{\rm \pi}\right\}[1+g(r/\delta)]\, {\rm d} a'=Q_\delta(a,t), \end{gather}

where $Q_\delta$ modifies $Q$ given in (3.12) by replacing $I$ with desingularized $I_\delta$ as follows:

(3.19)\begin{equation} I_\delta=\int\kern-8pt -_0^1 \frac{\gamma'{\rm \pi}(\dot{z}-\dot{z}')}{\sin^2 {\rm \pi}(z-z')}[1+g(r/\delta)]\, {\rm d} a'=\int_0^1 \left[\frac{\gamma'{\rm \pi}(\dot{z}-\dot{z}')}{\sin^2 {\rm \pi}(z-z')}+\frac{\gamma\dot{z}_aB}{{\rm \pi} z_a^2}\right][1+g(r/\delta)]\, {\rm d} a'. \end{equation}

3.1.3. Initial conditions

The initial values of $z$ and $\gamma$ considered here are derived from the same normal mode used in the linear theory (§ 2.5), where the perturbed interface shape is

(3.20)\begin{equation} z(a,0)=a+{\rm i}\epsilon \sin(2{\rm \pi} a).\end{equation}

Using the initial travelling wave ansatz $\hat {\eta }=\exp (-{\rm i}\varOmega t)$, as well as (2.9) and (3.1a,b), the corresponding circulation distribution at order $O(\epsilon )$ is

(3.21a,b)\begin{equation} \varGamma(a,0)=\frac{\epsilon \varOmega}{\rm \pi}[1-\cos(2{\rm \pi} a)], \quad \gamma(a,0)=2\epsilon \varOmega \sin(2{\rm \pi} a), \end{equation}

where we recall that

(3.22)\begin{equation} \varOmega=\sqrt{\frac{kA}{{Fr}^2}+\frac{k^3}{2{We}}}. \end{equation}

3.3.1. Amplified GCWs, $\eta _{max}$ and associated energy

First, we validate the linear theory for regimes I and IV of flows of short duration $T_2$ (or small $S_\infty$). We expect the amplitude of the interface in these flow regimes to sustain long term oscillations of decreasing amplitudes. We can compare our prediction of the maximum amplitude $\eta _{max}$, (2.14), with its nonlinear counterpart defined as $\eta _{max}=\max _{1>a\geq 0, t>0}|y(a,t)/\epsilon |$. Figure 11(a) shows $\eta _{max}$ for impulses of high and low strength $M$ (in regimes I and IV, respectively), where we find good agreement between the linear and nonlinear predictions of $\eta _{max}$. However, while maintaining a similar maximum displacement, the spatial wave shape given by the nonlinear theory starts to deviate from the linearly assumed harmonics as the flow action $S_\infty$ increases. This is shown in figure 11(b) for the large $M$ case for three impulses, where the nonlinear surface profile is shown when $\eta _{max}$ is reached.

Figure 11.

(a) Comparison of maximum interface amplitude $\eta _{max}$ as a function of flow action $S_\infty$, between the linear and nonlinear theories, for high and low impulse strength $M$. (b) Nonlinear interface profiles $z=x+iy$ at maximum amplitude obtained for large $M=14.6$ and $S_\infty =0.1, 0.15, 0.17$ (equivalently, increasing normalized impulse duration $T_2=0.05, 0.07, 0.08$, with $T_1=0.1T_2$). Although the linear theory captures the nonlinear maximum amplitude for the plotted range of $S_\infty$ well in (a), the associated wave shapes given in (b) are nonlinear and clearly deviate from the single harmonic assumed in the linear theory. The regimes covered in this figure are regimes I and IV as $M$ changes.

Next, we examine in detail the flow field, interface response and perturbation energy system given by two typical impulses: one with high effective strength $M=14.6$ and short normalized duration $T_2=0.07$ (regime I), and another with low strength with $M=1.46$ but longer duration with $T_2=1.5$ (regime II). For each of these two impulses, detailed flow streamlines and kinetic energy density are illustrated in figure 12 at progressive time instants. The streamlines are computed according to the flow velocity field $q(Z,t)$ given in (3.4). The energy density distribution $\varrho$ is defined here as $\varrho (Z,t)=\rho _j(|q(Z,t)|^2/2-U_j^2/2)$, where $j=1,2$ if $Y\lessgtr y$. In each fluid region, normalized $\bar {\varrho }=\varrho /\max _{Y\lessgtr y} |\varrho |$ is shown as a colour map, so that $-1\leq \bar {\varrho } \leq 1$ in both regions. The analysis that leads to the energy results shown here is given in Appendix C. To separate the different scaling used for the liquid and gas, the ratio between the two normalization factors, $r_k=\max _{Y> y} |\varrho |/\max _{Y< y} |\varrho |$ (gas over liquid), is also reported. In figures 12(a)–12(d), flow details are given for the low strength impulse with $M=1.46$ (regime II) at the initial time $T=0$, the impulse peak time $T=T_1=0.15$, the maximum interface amplitude time $T=T_m=0.8$ and the impulse stop time $T=T_2=1.5$. We see that, by the end of the flow impact, energy gains are concentrated around the interface, and because $r_k=0.001$ at this instance, most of the kinetic energy is indeed absorbed by the liquid phase. More interestingly, the flow field obtained for high strength impulse with $M=14.6$ (regime I) is depicted in figure 12(e)–12(h). Formation of opposite shearing, and/or vortex strength, along the interface here is kinematically responsible for the interfacial growth immediately after the flow impulse at $T=T_2=0.07$. At maximum perturbation time $T=T_m=0.3$, a finer vortex pattern is created, implying imminent interface distortion at smaller length scales, powered by a narrow encircling zone of high kinetic energy.

Figure 12.

Flow streamlines and perturbative kinetic energies (colour map) around the interface in response to a flow impulse of low strength $M=1.46$ in (a–d) (regime II), and high strength $M=14.6$ in (e–h) (regime I). The time instants are labelled in the $U_2$ history. The same initial condition given in (a) is shared between the two sequences. Normalized energy density $\bar {\varrho }$ in each fluid region at each time instance is given. Ratio between the two reference scales at each instance is $r_k=\max _{Y> y} |\varrho |/\max _{Y< y} |\varrho |$. Illustrations for the corresponding $U_2$ profiles are not to scale. The figure reveals the difference between peri-impulse amplification in (a)–(d) and post-impulse amplification in (e,f).

3.3.2. Transition to breaking waves

Further increasing the imposed flow duration necessarily generates large surface disturbances that cannot be supported by periodic, oscillatory waves. Instead, wave breaking could occur. For example, formation of a plunging breaker is captured in figure 13(a) for the high strength and moderate duration impulse with $M=14.6$ and $T_2=0.5$ (regime III), where the $y$-values are successively elevated for each interface profile at different times. Note that the numerical method for breaking waves suffers additional flow instabilities caused by the dispersive capillary waves propagating along the interface found in figure 13(a), and more importantly, the Rayleigh–Taylor instability due to reversed density stratification, thus we only use it to investigate onset of overturning in what follows.

Figure 13.

Wave breaking for flow impulse of $M=14.6$ and $T_2=0.5$, an example of a regime III response. Progressive interface profiles with elevated $y$ values by increasing constants are shown in (a). The corresponding time and impulse instants, $U_2(T)$, are marked in (b). Four close ups at $T=0.717, 0.721, 0.723, 0.726$ in (c) delineate the overturning, and in (d) the associated curvature increase is shown as a function of $T$. Evolution of the perturbative energies is given in (e) where the inset zooms in for $E_s$ and $E_p$. This is an example close to a classic KH instability that leads to interface roll-up.

The impulse velocity as a function of time is given in figure 13(b), where the wave shape corresponding times are marked. Clearly, dramatic interface rolling rapidly takes place over a relatively short time window. The accelerating rotation of wave shape $z$ overturning around the point of infinite slope at $T=0.0723$ is shown in the close-up in figure 13(c). During this process, although curvature singularity associated with the present vortex sheet is presumably removed by surface tension (Baker & Nachbin Reference Baker and Nachbin1998), substantial increase of $\kappa$ is still numerically observed during wave breaking in figure 13(d). Lastly, evolution of the perturbative energies discussed in § C.2 is shown in figure 13(e), where the liquid flow accumulates kinetic energy throughout the process while the gas flow consistently loses relative kinetic energy. The surface and potential energies of the interface also increase significantly compared with the system's initial condition.

Having established two categories of interface responses to a given flow impulse, namely amplified GCW and roll-ups, in figure 14 we investigate the stability transition as flow duration increases. Using low effective strength $M=1.46$, figure 14(a) shows four simulated flows of increasing $T_2=2, 2.4, 2.5, 3$ (regime II). Accordingly, the history of wave height, computed as $H(t)=\max _a(y(a,t))-\min _a(y(a,t))$, in response to the imposed impulses is shown in figure 14(b). It is clear that regime transition takes place between $T_2=2.4$ and $T_2=2.5$; the interface survives the entire short impulse without breaking and continues to oscillate after reaching peak height at $H=0.12$ while, for the slightly longer impulse, $H$ rises monotonically and develops a vertical slope that leads to a breaker at $T=1.27$. Note that wave breaking in this case occurs during the KH stable portion of the flow impulse, i.e. times when $MU_2<1$. To emphasize the importance of continued interaction between the flow impulse and the interface in sustaining the breaking process even after the conditions return to being classically KH stable again, we impose a hypothetical flow impulse that follows the $T_2=2.5$ curve until $T=1.18$ and drastically decays to zero at $T=1.24$ (see the dashed line with an open square in figure 14a). The resulting $H$ evolution given by the dashed curve in figure 14(b) suggests that a breaker does not develop and the wave amplitude starts declining if the impulse is rapidly turned off. Comparison between the interface profiles at peak amplitude given by $T_2=2, 2.4$ and those at turning point generated by $T_2=2.5, 3$ is shown in figure 14(c).

Figure 14.

Stability transition of interface response to flow impulses of increasing duration. Regime II cases leading to destabilization with $M=1.46$ are given in (a)–(c), with amplification occurring during the impulse's duration. Regime I to regime III with $M=14.6$ are shown in (d)–(f), with amplification occurring mostly post-impulse. Panels (a) and (d) plot the sequence of imposed flows $U_2(T)$, (b) and (e) give the resulting wave height $H(T)$. In (a) the horizontal dividing line marks linear stability. The triangles mark instants when the interface is at peak amplitude without wave breaking, whereas the asterisks mark the onset of wave overturning when a vertical slope occurs. The solid circles indicate times after which a larger numerical smoothing length $\delta =1/128$ is used. The open squares designate impulse end time of the hypothetically switched off impulse drawn using a dashed line. For each impulse, (c) and (f) display the wave shape at peak amplitude or breaking point. By gradually increasing the impulse duration $T_2$, the transition of interface stability from persisting waves into breaking waves is demonstrated.

Analogous results for the high strength flow impulses of $M=14.6$ and $T_2=0.06, 0.07, 0.08, 0.09$ are shown in figures 14(d)–14(f) (transitioning from regime I into III). The spontaneous amplitude growth immediately after the end of the impulse is clear for all impulses, except for the longest one with $T_2=0.9$, where instead a breaker forms near the end of the impulse, marked by the asterisk sign (figure 14d). Interestingly, figures 14(e) and 14(f) show that the wave steepness at turning point in this case, $H=0.09$, is lower than the peak height attainable by impulses of shorter duration, in the regime where post-impulse oscillation is observed. This breaking height of $H=0.09$ is also smaller than those seen in figure 14(b) for the weaker impulses. Nonetheless, all values of critical $H$ at which breakers emerge in figure 14 fall within the range surveyed by Perlin et al. (Reference Perlin, Choi and Tian2013). Further, wave breaking with $T_2=0.09$ here, albeit at a relatively smaller critical $H$, must be supported by a larger energy influx from the flow impulse. Indeed, figure 15 shows the total energy injection into the two-fluid system, as well as the kinetic energy gain in the liquid phase, produced by the entirety of the flow impulse, up to $T_2=0.08$ (see Appendix C for energy calculations). The matching linear and nonlinear results show exponential energy growth as $T_2$ increases that is predominantly in the form of liquid kinetic energy $E_{k_1}$. Therefore, regime transition to interface rolling comes as no surprise if $T_2$ is further increased. Finally, note that with the current numerical scheme where surface tension effect is not completely resolved due to smoothing, we do not establish the long-term stability of post-impulse surface waves of large amplitude (e.g. gravity waves of $H>0.14$). We illustrate such oscillatory behaviour of high frequency capillary waves and its numerical properties in Appendix B.

Figure 15.

Normalized total energy injection into the two-fluid system (circles), and the kinetic energy gain in the liquid phase (squares), generated by regime I flow impulses of increasing duration $T_2$. Results are obtained at $T=T_2$ after which $E$ is conserved, and $E_{k_1}$ becomes the total kinetic energy in the liquid phase. This also show goods agreement with the linear theory from § C.1.

3.3.3. Breaking due to transient growth

So far, the initial perturbation size $\epsilon =0.001$ in (3.20) and (3.21a,b) has been used, although the phenomenon of over-energized interface losing stability is not limited to such specific choice of initial conditions. In fact, here we show that even transient growth supported by a classically KH stable flow, as introduced in § 2.5, can cause transition toward wave breaking.

Figure 16 shows two impulses of varied duration $T_2=1, 2$, but the same strength $M=0.95<1$ (regime II). Linear analysis shows that a maximum interface amplification exists for transient growth when $M<1$, independent of the imposed flow duration, and in this case $\eta _{max}\lesssim O(10)$ according to (2.19). Therefore, a larger perturbation size $\epsilon =0.05$ is adopted to promote wave destabilization. As a result, evolution of wave steepness $H$ follows in figure 16(b), where the short impulse yields stable oscillations whereas the long impulse causes the wave to overturn, starting at $T=0.534$ marked by the asterisk. The nonlinear solutions here are also compared with the linear ones, where the results match qualitatively for the short impulse, with particularly good agreement in the early stages. However, for the long impulse, the two predictions diverge around the time of peak $H$ given by the dashed line, after which oscillation continues for one and wave breaking initiates for the other. Figure 16(c) shows a sequence of wave profiles taken at times marked by open circles in figure 16(a,b), and starting at $T=0$. Clearly, the initial perturbation wave travels in the flow direction while the wavefront keeps steepening, until overturning occurs drastically over a short window of time. Note that the entire breaking process remains in the flow regime where the interface is linearly classically KH stable.

Figure 16.

Wave breaking during transient growth despite the flow regime being classically KH stable. (a) Two impulses $U_2(T)$ with weak effective strength $M=0.95$ are applied. (b) Comparison between the linear (dashed lines) and nonlinear (solid lines) predictions for wave height $H$ driven by the two flow impulses as a function of time. (c) Snapshots of wave shapes from the initial condition $\epsilon =0.05$, to the formation of wave breaker. The corresponding times are marked using open circles in (a) and (b). This is an example of wave breaking in regime II induced by increasing the duration of a classically KH stable flow impulse.

To understand this surprising behaviour, insights are gained from § A.2, where (A2) shows that the asymptotic solution of the linear system for small flow strength $M$, where $H=2|\eta | \epsilon$ is controlled by the convolution integral between flow action $U_2^2$ and the unforced interface GCW of the form $\sin (4{\rm \pi} T)$. The GCW oscillation offers a finite time scale for the imposed flow to resonate with the interface, so the wave amplitude grows. In the small $M$ limit, this window is exactly $T\in (0,1/4)$ according to (A2). However, for larger $M<1$, this time of overlap can be longer as observed in the linear solutions shown in figure 16(b). Further, figure 16 shows that, with sufficient amplification during the transient flow–interface resonance, wave breaking can occur, even if the flow is classically KH linearly stable all along. The transition from persisting waves to breaking waves is further demonstrated in figure 17 for weak impulses ($M=0.95$) of increasing duration $T_2$. If the amplified wave persists after reaching peak amplitude, we can define a maximum wave steepness $H_{max}=\max _{0\leq T \leq 1}H(T)$ from each airflow impulse. In this regime, our linear analysis captures well that $H_{max}$ is an increasing function of $T_2$. However, as $T_2$ increases, the interface response deviates increasingly from persistent waves (stable over at least $T\leq 1$) to breaking waves, where the wave shape overturns. For these long impulses of $T_2>1.3$, $H_\infty =H(T^*)$ is obtained at the overturning time, $T^*=\min \{T>0:\, {{\rm d} y}(a,T)/{{\rm d}\kern0.7pt x}(a,T) = \infty \}$, and is given in figure 16 instead of $H_{max}$. The value of $H_\infty$ appears to plateau for large $T_2$, suggesting a ceiling for wave amplitude growth fuelled by adequate cumulative exposure of the interface to the impulse during transient resonance, which, in turn, induces wave breaking.

Figure 17.

Maximum wave steepness $H_{max}$ (circles) and overturning wave steepness $H_\infty$ (triangles) as functions of impulse duration $T_2$. The dashed line gives $H_{max}=2\epsilon |\eta |$ from linear theory. The sharp transition between waves for small $T_2$ and breakers for large $T_2$ is highlighted. Constant $M=0.95$ and $\epsilon =0.05$ are used, while the range of $T_2$ here spans flow regimes IV to II, ultimately causing destabilization.