Effects of surface tension on the Richtmyer-Meshkov instability in fully compressible and inviscid fluids

Novel numerical simulations investigating the Richtmyer-Meshkov instability (RMI) with surface tension are presented. We solve the two-phase compressible Euler equation with surface tension and interface reconstruction by a volume-of-fluid method. We validate and bridge existing theoretical models of effects of surface tension on the RMI in the linear, transitional, and nonlinear postshock growth regimes. After deriving a consistent nondimensional formulation from an existing linear incompressible theory predicting perturbation development under large surface tension, we find good agreement with theoretical prediction in the small-amplitude (linear) oscillatory regime for positive Atwood numbers, and we show that negative Atwood numbers can be accommodated by an appropriate modification to the theory. Next, we show good agreement with nonlinear theory for asymptotic interface growth in the limit of small surface tension. Finally, we use the nondimensional formulation to define a heuristic criterion which identifies the transition from the linear regime to the nonlinear regime at intermediate surface tension. These results highlight the utility of this numerical method for compressible problems featuring surface tension, and they pave the way for a broader investigation into mixed compressible/incompressible problems.

Figure 1

Sketches for the pre- (a) and postshock (b) states of the compressible RMI problem with surface tension. The domain is separated by a sinusoidal interface featuring initial amplitude ${\eta }_0^{-}$ and wavelength $\lambda$. The fluids to the left and right of the interface are labeled “1” and “2,” respectively. The incident shock with Mach number $M_{s,I}$ changes the state of fluid 1 from preshock (with superscript “$-$”) to intermediate (with subscript “L”) before its interaction with the interface, which compresses the interfacial perturbation to ${\eta }_0^{+}$. The transmitted shock and reflected wave bring the two fluids to their final postshock states (with superscript “$+$”), while the interface perturbation develops under the influence of surface tension $\sigma$.

Figure 2

The development of postshock density values ${\rho }_i^{+}$ in cases with preshock Atwood number $A^{-}=9/11$ (a) and $A^{-}=-9/11$ (b) measured from numerical diagnostics. The postshock densities of the two fluids are observed to eventually settle down at steady-state values, which are chosen as the postshock values determined by numerical diagnostics.

Figure 3

Numerical results compared with Mikaelian's theory [5]. Upper row: strong shock ($M_{s,I}=2$) with negative Atwood number ($A^{-}=-9/11$); middle row: strong shock ($M_{s,I}=2$) with positive Atwood number ($A^{-}=9/11$); lower row: weak shock ($M_{s,I}=1.2$) with positive Atwood number ($A^{-}=9/11$). Left column: perturbation growth scaled with the natural units. Right column: results scaled using the postshock dimensionless parameters. Good collapsing patterns are observed for strong-shock cases $(M_{s,I}=2)$, among which those with positive Atwood number $A^{-}=9/11$ show a good qualitative agreement with Mikaelian's theory [5], whereas the weak-shock cases display poor collapsing under the normalization.

Figure 4

Upper row: Comparison of the simulation results with the theortical predictions of Richtmyer [24] and Vandenboomgaerde et al. [22]. Lower row: simulation results scaled using the modified model in cases with surface tension. (a, c) $A^{-}=9/11$, (b, d) $A^{-}=-9/11$. The theory of Vandenboomgaerde [22] shows a better match with simulation results in both negative and Atwood number cases without surface tension, leading to improved performance of the modified theoretical model.

Figure 5

A series of ${\text{We}}^{-}=3800$ simulation snapshots showing the evolution of the perturbed interface from the initially sinusoidal shape to one with a bubble (across the periodic boundaries) and a spike (at the center).

Figure 6

Development of bubble (a) and spike (b) velocity for transitional and nonlinear cases with $A^{-}=9/11$. As Weber number increases, the curbing effect of surface tension on the postshock perturbation growth diminishes, and the time development patterns of bubble and spike velocities remain roughly the same within a large range of high Weber numbers (greater than ${10}^3$).

Figure 7

Comparison between simulated (solid lines) and theoretical (dashed line) bubble velocity developments in cases with ${\text{We}}^{-}=480,1900,3800,$ and 5700. Good agreements with Sohn's theory are found at asymptotically large Weber numbers.

Figure 8

Normalized perturbation development curves for $A^{-}=9/11$. (a) $s^{-}=0.02\pi$; (b) $s^{-}=0.03\pi$; (c) $s^{-}=0.04\pi$; (d) comparisons of cases with $s^{-}=0.02\pi ,0.03\pi$, and $0.04\pi$, at ${\text{We}}^{+}$ values around 90. For panels (a–c), the zoom-in views near the first peaks of the curves are displayed, and the peak points of each curve are marked with circles. Rightward shift of the peak points is found as the nonlinear indicator.

Figure 9

Left: phase diagram showing simulation cases in linear growth-rate regime (squares) and transitional regime (crosses). The boundary between the two is selected as ${\chi }_{1.8}=0.06$ and plotted in dashed line, while ${\chi }_1=0.6$ is also plotted in dotted line for comparison. Right: relationship between $\mathrm{\Delta }$ and ${\chi }_{1.8}$ with different $s^{-}$ setups. A common growth stage of $\mathrm{\Delta }$ is reached for different $s^{-}$ setups as ${\chi }_{1.8}$ increases, and the critical ${\mathrm{\Delta }}_c$ and ${\chi }_{1.8}$ values are plotted.

Figure 10

Convergence test results (Upper row: heavy-light density setup without (a) and with (b) surface tension; lower row: light-heavy density setup without (c) and with (d) surface tension). Good numerical convergence is observed for all the inviscid test cases.