Growth of liquid-gas interfacial perturbations driven by acoustic waves

Diagnostic ultrasound has been shown to cause lung hemorrhage in a variety of mammals, though the underlying damage mechanisms are still unclear. Motivated by this problem, we use numerical simulations to investigate the interaction of an ultrasound wave with the alveolar tissue-air interface. A planar, single-cycle, trapezoidal waveform propagates in tissue (modeled as water) and impinges upon an alveolus of the lung (modeled as air); to represent the alveolar surface roughness, the interface consists of a small-amplitude single-mode perturbation. Because of the sharp density gradient at the interface, we hypothesize that ultrasound waves, despite their relatively low amplitude, deposit sufficient baroclinic vorticity to drive perturbation growth. Our simulations show that the perturbation amplitude grows to sizes many times larger than the original value, well after the wave has passed. We demonstrate that conventional (linear) acoustics cannot account for such deformations; instead, the perturbation growth is driven by nonlinear effects: the baroclinic vorticity deposited along the interface, due to the misalignment of the pressure gradient (across the wave) and the density gradient (across the perturbed gas-liquid interface). Based on dimensional analysis and scaling, we observe that the perturbation amplitude and length of the interface scale with the circulation density and grow according to power laws in time. If the time interval between the pressure increase and decrease is sufficient, both deposit vorticity of the same sign, thus enhancing the perturbation growth; conversely, if the interval is too short, the vorticity deposited by the pressure increase is canceled by the decrease. A further consequence is that one may be able to control the growth of such perturbed interfaces by modulating the incoming waveform.

Figure 1

(a) Schematic description of the physical problem of interest (ultrasound pulse in soft tissue impinging upon the first alveolus it encounters). (b) Computational setup of the model problem (acoustic wave in water impinging upon a sinusoidally perturbed air interface of initial amplitude $a_0$).

Figure 2

Ideological progression from ultrasound pulse and shock to our baseline trapezoidal wave, which can be analyzed with Richtmyer-Meshkov-inspired analysis.

Figure 3

Density contours at $t/(\ell /c)=4.75$, 47.5, 475, and 1424 for the baseline $p_a=10$ MPa, $L=45\ell$ trapezoidal wave. The black solid line is the $\alpha =0.5$ volume fraction isoline and the black dashed line the initial mean interface location.

Figure 4

Interface perturbation amplitude history $a\left(t\right)/a_0$ for the baseline $p_a=10$ MPa, $L=45\ell$ trapezoidal wave, for (a) $t/(\ell /c)\le 120$ and (b) $t/(\ell /c)\le 5000$. In (a), times at which different stages of the incoming trapezoidal pressure wave first encounter the interface are indicated as $t_1$ (the compression), $t_2$ (the constant elevated pressure $p_a$), $t_3$ (the rarefaction), and $t_4$ (the return to ambient pressure).

Figure 5

The $y$ locations of the bubble and spike for the baseline $p_a=10$ MPa, $L=45\ell$ trapezoidal wave, during and shortly after the wave-interface interaction. By definition, the bubble is the top (blue solid) curve and the spike is the bottom (red dashed) curve. The black dotted line shows $t_{\mathrm{\Gamma }}$, as defined by the minimum bubble location.

Figure 6

Vorticity contours at $t/(\ell /c)=4.75$, 47.5, 475, and 1424 for the baseline $p_a=10$ MPa, $L=45\ell$ trapezoidal wave case. The black solid line is the $\alpha =0.5$ volume fraction isoline and the black dashed line is the initial mean interface location.

Figure 7

Cumulative vorticity (in $y$) along the interface, ${\int }_{-\infty }^{+\infty }\tilde{\omega }(x,y)dy$, where $\tilde{\omega }=\omega$ for $0<\alpha <1$ and is otherwise zero (in pure water or air), at $t/(\ell /c)=4.75$ (blue solid line), 47.5 (red dashed line), 475 (green dotted line), and 1424 (purple dash-dotted line).

Figure 8

Circulation history for the right-half domain for the baseline $p_a=10$ MPa, $L=45\ell$ trapezoidal wave case, for (a) $t/(\ell /c)\le 120$ and (b) $t/(\ell /c)\le 5000$. In (a), times at which different stages of the incoming trapezoidal pressure wave first encounter the interface are indicated as $t_1$ (the compression), $t_2$ (the constant elevated pressure $p_a$), $t_3$ (the rarefaction), and $t_4$ (the return to ambient pressure).

Figure 9

Histories of the (a) interface amplitude and (b) circulation for trapezoidal wave cases with $L=45\ell$ and $p_a=5.0$ (blue solid line), 7.5 (red dashed line), 10.0 (green dotted line), and 12.5 (purple dash-dotted line) MPa, for $t/(\ell /c)\le 5000$.

Figure 10

Interface amplitude scaled by circulation density at $t_{\mathrm{\Gamma }}$ for the trapezoidal wave cases with $L=45\ell$ and $p_a=5$ (blue solid line), 7.5 (red dashed line), 10 (green dotted line), and 12.5 (purple dash-dotted line) MPa. Time is synchronized based on phase inversion. The black dashed line shows the power-law growth as $t^{3/5}$.

Figure 11

Histories of the (a) interfacial length and (b) scaled interfacial length for the trapezoidal wave cases with $L=45\ell$ and $p_a=5$ (blue solid line), 7.5 (red dashed line), 10 (green dotted line), and 12.5 (purple dash-dotted line) MPa. Time is synchronized based on phase inversion. The black dashed line shows the power-law growth as $t^{1/2}$.

Figure 12

(a) Interface amplitude and (b) circulation histories for the $p_a=10$ MPa trapezoidal wave case with $L=45\ell$ (blue solid line), $35\ell$ (red dashed line), $30\ell$ (green dotted line), $20\ell$ (purple dash-dotted line), $15\ell$ (orange line with circles), and $10\ell$ (yellow line with triangles).

Figure 13

(a) Interface amplitude and (b) circulation histories for the $p_a=+10$ (blue solid line) and $-10$ (red dashed line) MPa trapezoidal wave cases with $L=45\ell$.