Numerical Study of Amplitude-Driven Flow Dynamics in Shocked Heavy-Fluid Layers

Abstract

In this study, a comprehensive numerical investigation of amplitude-driven flow dynamics in shocked heavy-fluid layers is presented to focus on the evolution of the Richtmyer–Meshkov instability (RMI). A high-order mixed local discontinuous Galerkin scheme is employed to resolve the complex interactions between shock waves and perturbed interfaces within a compressible viscous flow framework. Impacts of the initial interface amplitudes are systematically examined through a series of single-mode configurations with amplitude–wavelength ratios ranging from $a_0/\lambda =0.025$ to $0.4$. The simulations capture the complete transition from early linear growth to nonlinear roll-up and subsequent mixing. This investigation illustrates that increasing the initial perturbation amplitude enhances baroclinic vorticity generation, intensifies interfacial deformation, and accelerates the onset of secondary instabilities. Low-amplitude interfaces maintain nearly symmetric deformation with delayed nonlinear transition, whereas high-amplitude cases exhibit pronounced spike–bubble asymmetry, stronger curvature, and rapid Kelvin–Helmholtz roll-ups. Quantitative diagnostics of the circulation, enstrophy, and kinetic energy demonstrate that both baroclinic torque and mixing intensity scale directly with the initial perturbation amplitude. This study offers new physical insight into amplitude-dependent shock–interface interactions and elucidates the mechanisms governing vorticity amplification and energy redistribution in RMI flows.

1. Introduction

Shock-driven flow instabilities in stratified or layered fluids have long been recognized as central phenomena in compressible fluid dynamics, with implications across a wide range of physical systems [1,2]. The passage of a shock across a disturbed density interface produces baroclinic vorticity through the non-alignment of pressure and density gradients, leading to intricate flow dynamics marked by interface deformation, vortex development, and ensuing mixing. This process underlies several classical instability mechanisms—most notably the Richtmyer–Meshkov instability (RMI) [3,4]—which serves as the impulsive counterpart to the continuous acceleration-driven Rayleigh–Taylor instability (RTI) [5,6]. Together, these instabilities govern the onset and growth of turbulence and mixing in many shocked or accelerated multiphase flows. Over recent decades, the RMI has attracted sustained attention through analytical, experimental, and computational studies, reflecting its fundamental role and wide-ranging significance across scientific and engineering applications. In inertial confinement fusion (ICF), the control of RMI-induced perturbation growth is essential for achieving symmetric target compression and maximizing energy yield [7,8]. Similarly, the RMI plays a critical role in astrophysical phenomena, such as supernova explosions, where shock–interface interactions influence the large-scale morphology of ejecta [9,10]. In aerodynamic and combustion systems, including scramjets and shock–flame interactions, the mixing enhancement driven by the RMI can improve combustion efficiency and stability [11]. Because of its widespread occurrence and inherent multi-scale characteristics, the detailed investigation of the RMI continues to be a key focus within compressible flow research.

Baroclinic vorticity generation in compressible flows arises fundamentally from the misalignment of pressure and density gradients and is not restricted to impulsive shock–interface interactions alone. Classical theoretical studies by Crocco [12] and Hayes [13] established the role of thermodynamic non-uniformity in producing vorticity in compressible gases, providing a foundation for later analyses of shock-driven flows. Subsequent investigations demonstrated that both discontinuous and smooth pressure variations can generate vorticity through sustained baroclinic torque. In particular, Kevlahan [14] showed that interactions between shock or acoustic waves and density inhomogeneities lead to significant vorticity deposition, even in the absence of sharp interfaces. Extending this concept, Povitsky and Ofengeim [15] numerically demonstrated that gradual pressure variations associated with Prandtl–Meyer expansion waves can also produce baroclinic vorticity when interacting with vortical density inhomogeneities, highlighting the importance of continuous pressure–density misalignment in compressible flows. Within the context of impulsive acceleration, Richtmyer [3] and Meshkov [4] identified baroclinic torque as the dominant mechanism driving interface instability when a shock wave interacts with a perturbed density interface, laying the groundwork for modern studies of the RMI.

The interaction of a shock wave with a perturbed material interface has long served as a canonical problem in compressible flow dynamics, with early studies elucidating the nonlinear growth behavior of the RMI and the fundamental role of baroclinic vorticity generation arising from pressure–density misalignment at shocked interfaces. Velikovich and Dimonte [16] developed a high-order single-mode nonlinear perturbation theory to analyze the RMI at an impulsively accelerated incompressible interface subject to constant supporting pressure. Subsequently, Nishihara et al. [17] presented a unified theoretical framework for linear and nonlinear RMI, deriving analytical expressions for asymptotic growth rates and nonlinear saturation based on circulation dynamics, and demonstrating vorticity generation in fluids, solids, and liquids under shock-driven perturbations. Sadot et al. [18] conducted shock-tube experiments to validate theoretical predictions of shock-induced RMI, examining single-mode bubble–spike evolution and two-bubble interactions during both early and late nonlinear stages. Collins and Jacobs [19] employed planar laser-induced fluorescence (PLIF) in membrane-free shock-tube experiments to visualize and quantify the RMI at an air/SF6 interface, demonstrating excellent agreement of early-time growth with linear theory and late-time evolution with nonlinear models, as well as enhanced turbulent mixing at higher shock strengths. Jacobs et al. [20] performed shock-induced experiments on the late-time evolution of single-mode RMI, revealing turbulence development and vortex-core breakup under stronger shocks. Subsequently, Jourdan and Houas [21] experimentally investigated single-mode RMI across light/heavy, close-density, and heavy/light gas interfaces at low Mach numbers. The study demonstrated nonlinear growth-rate reduction relative to linear theory and good agreement with reduced-growth nonlinear models for different density configurations. Gorodnichev et al. [22] analyzed an RMI arising from high-velocity impact between solid plates with initially non-uniform density fields, developing linear and nonlinear analytical models that identify distinct disturbance regimes. Wright et al. [23] systematically investigated the influence of fluid adiabatic indices on a strong-shock-driven RMI using smooth particle hydrodynamics, demonstrating their impact on post-shock interface velocity, early-time growth rates, and interfacial mixing. Most of these studies focused on the idealized configuration of a semi-infinite single-mode interface, which serves as a fundamental model for isolating baroclinic vorticity deposition and interfacial deformation mechanisms. In contrast, many practical systems involve finite-thickness fluid layers rather than isolated interfaces. In ICF, for instance, the imploding target consists of multiple material layers—such as an ablator, a solid deuterium–tritium shell, and a gaseous core—where shock-induced interfacial instabilities and layer coupling can degrade compression symmetry and reduce fusion yield [24,25]. These examples underscore the importance of understanding the RMI in finite-thickness configurations, where interfacial coupling and multi-layer interactions fundamentally alter the instability dynamics [26,27,28].

Investigating the evolution of shocked fluid layers poses a formidable challenge owing to the complex interplay between reverberating shock waves and the coupled dynamics of multiple interfaces. Over the past few decades, extensive theoretical efforts have been devoted to elucidating these interactions in finite-thickness configurations. The pioneering work of Taylor [6] first examined the RTI in a confined liquid layer, revealing how interface coupling becomes increasingly pronounced as the layer thickness decreases. Building upon this foundation, Mikaelian [29,30,31,32] derived linear analytical solutions describing the evolution of stratified fluids subjected to both the RTI and RMI. Subsequently, Jacobs et al. [33] developed a linear model to predict perturbation growth rates and a vortex model to describe the evolution of the mixing width in shocked heavy-fluid layers, which was later validated experimentally by Prestridge et al. [34] through circulation measurements in a shocked gas curtain.

While these studies primarily focused on interfacial coupling effects during early-time evolution, more recent investigations have sought to refine predictive modeling under broader flow conditions. Liang and Luo [35] proposed an improved theoretical framework for gas-layer growth during the linear regime, accounting for rarefaction and transverse-wave effects at low Mach numbers. Gowardhan et al. [36] performed implicit large-eddy simulations of Mach 1.2 shock interaction with an ${\mathrm{SF}}_6$ gas curtain, examining the influence of initial conditions on material mixing and the transition to turbulence. Li. et al. [37] numerically investigated the interaction between a converging cylindrical shock and double density interfaces in the presence of a saddle magnetic field within the framework of ideal magnetohydrodynamics. Chen et al. [38] investigated, numerically and theoretically, the RMI of a light fluid layer driven by reflected shocks, revealing features of its interfacial evolution under reshock. Xie et al. [39] conducted a neural network-based numerical investigation on the growth rate of shocked heavy gas layers by varying the Mach number, Atwood number, thickness-to-wavelength ratio, and initial amplitude ratio, providing predictive models for upstream and downstream interface evolution. More recently, Singh [40] conducted computational studies of a shock-accelerated V-shaped heavy-fluid layer, demonstrating agreement with experimental observations and identifying Mach-number-driven transitions from initial linear distortion to nonlinear vortex formation and turbulent flow. Msmali et al. [41] reported high-resolution numerical investigations of a shock-driven single-mode RMI in a helium layer bounded by two interfaces and embedded within a nitrogen environment. Alsaeed et al. [42] employed a high-order method framework to study a shock-induced RMI in single-mode heavy-fluid layers, demonstrating the effects of different Atwood numbers on interfacial development, vorticity generation, and mixing characteristics. Subsequently, Alsaeed et al. [43] carried out numerical studies with a high-order method to assess the effect of heavy-layer thickness on shock-driven interfacial instabilities, showing how acoustic transit time, baroclinic circulation development, and interface morphology change for thin, intermediate, and thick configurations.

From a physical perspective, the impact of a planar shock on a disturbed heavy-fluid layer leads to partial transmission and reflection at each interface, producing intricate wave interactions and inducing baroclinic vorticity of opposite polarity at the leading and trailing interfaces [44]. The resulting vorticity fields interact in ways that are strongly dependent on the initial perturbation amplitude—constructively in some cases, enhancing interfacial roll-up and possibly inducing secondary shock focusing, or destructively in others, leading to partial wave cancellation and perturbation freeze-out [45]. To quantify these effects, amplitude-parameterized numerical experiments are essential. By systematically varying the initial amplitude while keeping other parameters fixed, one can identify the transition between coupled and decoupled interfacial regimes and track how shock-driven perturbations evolve temporally—from linear growth to nonlinear saturation—while energy and circulation are redistributed within the layer. Such analyses yield valuable insight into amplitude-controlled stabilization and destabilization mechanisms, providing pathways to regulate interface morphology and mixing behavior in practical shock-driven systems.

This study employs high-order discontinuous Galerkin simulations to resolve amplitude-driven shock–interface dynamics in a heavy-fluid layer with reduced numerical dissipation. A mixed local discontinuous Galerkin (LDG) framework is used to accurately capture steep gradients and small-scale vortical structures. The methodology introduces a systematic decomposition of vorticity generation into baroclinic, viscous, and dilatational contributions, together with a unified post-processing framework for circulation, enstrophy, and kinetic-energy diagnostics. A controlled amplitude-scaling strategy is further adopted to distinguish linear-stage growth from nonlinear amplification, providing clear insight into how initial interface amplitude governs the transition to nonlinear mixing.

The paper is structured to present a coherent progression from the physical formulation to the interpretation of numerical findings. Section 2 outlines the problem setup and the governing equations for the compressible multicomponent flow simulations. Section 3 details the adopted high-order numerical framework and validations. In Section 4, the results are analyzed, with an emphasis on amplitude-dependent interface evolution, circulation dynamics, and energy transfer. Section 5 concludes the study by summarizing the principal outcomes, discussing their physical significance, and indicating potential avenues for future research.

2. Problem Setup and Mathematical Formulation

2.1. Problem Setup

Figure 1 displays the two-dimensional configuration adopted to investigate the amplitude-driven dynamics in a shocked heavy-fluid layer. The numerical simulations are carried out in a rectangular domain spanning $[0,100]\times [0,200]{\mathrm{mm}}^2$ in the $(x,y)$ coordinate system. A heavy SF₆ gas layer with a finite thickness of $d=30\mathrm{mm}$ is placed between regions of lighter N₂ gas, establishing a standard light–heavy–light layered configuration. The layer is bounded upstream by a sinusoidally perturbed interface $I_1$ and downstream by a planar interface $I_2$. The upstream interface $I_1$ is initialized with a single-mode sinusoidal perturbation defined as

$$x=a_{0}cos\left({\displaystyle \frac{2\pi y}{\lambda }}\right).$$

(1)

Here, $a_0$ represents the initial amplitude of the perturbation, while $\lambda$ corresponds to its wavelength. The wavelength is fixed at $\lambda =100\mathrm{mm}$, while $a_0$ is systematically varied to examine its influence on the interfacial dynamics. The downstream interface $I_2$ remains planar, providing a reference surface for assessing transmitted and reflected wave interactions within the heavy layer.

A planar incident shock (IS) is introduced in the surrounding light gas and travels along the positive x-direction. At the initial time $t=0$, the shock front is located 25 mm from the left boundary, and the leftmost crest of the perturbed interface $I_1$ is positioned 5 mm behind the shock front. This arrangement ensures that the shock first interacts with the sinusoidal interface, triggering baroclinic vorticity deposition and subsequent RMI growth. In all simulations, the shock strength is maintained at a constant Mach number of $M_s=1.25$ in order to specifically examine the influence of the initial perturbation amplitude on the instability evolution. Five cases are considered with increasing initial amplitude–wavelength ratios ($a_0/\lambda$), as listed in

Table 1. Configuration of the shocked heavy-fluid layer adopted for the amplitude-dependent numerical simulations. Each case corresponds to a different initial perturbation amplitude $a_0$, while the wavelength $\lambda$ is kept constant.

Case	I	II	III	IV	V
$a_0$ (mm)	2.5	5	10	20	40
$a_0/\lambda$	0.025	0.05	0.1	0.2	0.4

. This systematic variation enables a quantitative assessment of how the initial interface amplitude governs early-time vorticity generation, interface deformation, and the nonlinear transition of the heavy-fluid layer under shock acceleration.

The numerical values reported in Table 1 are chosen to systematically investigate the effect of the initial perturbation amplitude on shock-driven instability evolution. The amplitude–wavelength ratios $(a_0/\lambda )$ are selected to cover the linear ($a_0/\lambda \le 0.05$), transitional ($a_0/\lambda \approx 0.1$), and nonlinear ($a_0/\lambda \ge 0.2$) regimes, consistent with classifications commonly adopted in previous RMI studies [46]. All other parameters are held fixed to isolate amplitude effects and enable direct comparison across cases.

The density difference between the two gases is quantified using the Atwood number,

$$At={\displaystyle \frac{{\rho }_h-{\rho }_l}{{\rho }_h+{\rho }_l}},$$

(2)

where ${\rho }_h$ and ${\rho }_l$ correspond to the densities of the heavy gas (SF₆) and the light gas (N₂), respectively. In this configuration, $At=0.66>0$, indicating that the heavier gas layer is embedded between lighter ambient regions. This positive Atwood number represents a dynamically unstable stratification conducive to studying amplitude-dependent evolution in shock-driven interfacial instabilities.

Table 2. Thermophysical parameters of the gas pair considered in this work, evaluated at the reference conditions $P_0=101,325\mathrm{Pa}$ and $T_0=293\mathrm{K}$.

Gas	Heat Ratio ($\mathit{\gamma }$)	Density ($\mathbf{g}{\mathbf{cm}}^{-3}$)	Specific Heat ($\mathbf{kJ}{\mathbf{g}}^{-1}{\mathbf{K}}^{-1}$)	Molecular Weight ($\mathbf{g}{\mathbf{mol}}^{-1}$)
N2	1.40	$1.25\times {10}^{-3}$	$1.04\times {10}^{-3}$	28.0134
SF6	1.09	$6.03\times {10}^{-3}$	$0.656\times {10}^{-3}$	128.491

lists the thermophysical characteristics of the gas mixture employed in the simulations, including the specific heat ratio ($\gamma$), density ($\rho$), specific heat at constant pressure ($C_p$), and molecular weight (M), all specified at the initial reference conditions.

2.2. Governing Equations

The governing framework for the amplitude-controlled shock–interface interaction in the heavy-fluid layer consists of the two-dimensional unsteady compressible Navier–Stokes–Fourier (NSF) equations formulated for an ideal non-reactive multicomponent gas mixture [47,48]. Within this formulation, mass, momentum, and total energy are conserved, with additional terms included to represent viscous dissipation and thermal transport relevant to shock-induced interfacial deformation and mixing. The conserved-state vector is defined as

$$\mathbf{U}={\left[\rho ,\rho \mathbf{u},\rho E,\rho Y_k\right]}^T$$

(3)

where $\rho$ denotes the fluid density, $\mathbf{u}$ the velocity vector, E the total specific energy, and $Y_k$ the mass fraction of the k-th species.

The governing equations can then be compactly expressed as a system of conservation laws,

$${\displaystyle \frac{\partial \mathbf{U}}{\partial t}}+\nabla ·{\mathbf{F}}_{\mathrm{c}}\left(\mathbf{U}\right)+\nabla ·{\mathbf{F}}_{\mathrm{d}}(\mathbf{U},\mathrm{\Pi },\mathbf{Q})=0,$$

(4)

where ${\mathbf{F}}_{\mathrm{c}}$ and ${\mathbf{F}}_{\mathrm{d}}$ stand for the convective and diffusive flux components, respectively. These flux components are given by

$${\mathbf{F}}_{\mathrm{c}}={\left[\rho \mathbf{u},\rho \mathbf{u}\mathbf{u}+p\mathbf{I},(\rho E+p)\mathbf{u},\rho Y_k\mathbf{u}\right]}^T,$$

(5)

$${\mathbf{F}}_{\mathrm{d}}={\left[0,\mathrm{\Pi },\mathrm{\Pi }·\mathbf{u}+\mathbf{Q},\rho D_k\nabla Y_k\right]}^T.$$

(6)

Here, p represents the thermodynamic pressure, $\mathbf{I}$ denotes the identity tensor, and $D_k$ is the diffusion coefficient associated with the k-th species. The viscous stress tensor $\mathrm{\Pi }$ and the heat flux vector $\mathbf{Q}$ are described using Newtonian viscosity and Fourier heat-conduction relations, respectively. For this purpose, the deviatoric component of the strain-rate tensor is introduced.

$$\mathbf{S}={\displaystyle \frac{1}{2}}\left(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^T\right)-{\displaystyle \frac{1}{3}}(\nabla ·\mathbf{u})\mathbf{I},$$

(7)

and express the stress and heat flux as

$$\mathrm{\Pi }=-2\mu \mathbf{S},\mathbf{Q}=-\kappa \nabla T,$$

(8)

where $\mu$ and $\kappa$ represent the dynamic viscosity and thermal conductivity of the gas mixture, respectively, and T denotes the temperature. The mixture composition is described via the mass fraction, defined as $Y_k={\rho }_k/\rho$. For a two-species system, one may simply set $Y_1=1-Y_2$. The total energy density is written as

$$\rho E={\displaystyle \frac{p}{\overline{\gamma }-1}}+{\displaystyle \frac{1}{2}}\rho {\left|\mathbf{u}\right|}^2,$$

(9)

where $\overline{\gamma }$ denotes the effective ratio of specific heats of the mixture. This quantity is obtained from the mixture-averaged specific heat capacities at constant pressure and constant volume, namely

$${\overline{C}}_p=\sum _k{z}_k{C}_{p,k},{\overline{C}}_v=\sum _k{z}_k{C}_{v,k},\overline{\gamma }={\displaystyle \frac{{\overline{C}}_p}{{\overline{C}}_v}},$$

(10)

with $z_k=Y_k/M_k$ representing the normalized mass fraction of species k. The individual species heat capacities follow the ideal-gas relations

$$C_{p,k}={\displaystyle \frac{{\gamma }_k{R}_u}{{\gamma }_k-1}},C_{v,k}=C_{p,k}-R_u,$$

(11)

where $R_u$ denotes the universal gas constant. The mixture pressure is obtained using Dalton’s law such that $p={\sum }_k{p}_k$ with $p_k={\rho }_k{R}_{k}T$. The transport properties of the mixture are computed through mass-weighted averaging; i.e.,

$$\left(\mu ,\kappa \right)=\left(\overline{\mu },\overline{\kappa }\right)\left({\displaystyle \frac{{\sum }_k{\mu }_k{Y}_k{M}_k^{-1/2}}{{\sum }_k{Y}_k{M}_k^{-1/2}}},{\displaystyle \frac{{\sum }_k{\kappa }_k{Y}_k{M}_k^{-1/2}}{{\sum }_k{Y}_k{M}_k^{-1/2}}}\right).$$

(12)

The viscosity of each species is determined using a Chapman–Enskog formulation [49],

$${\mu }_k=C_{\mu }{\displaystyle \frac{\sqrt{M_{k}T}}{{\Omega }_{\mu ,k}{\sigma }_k^2}},C_{\mu }=2.6693\times {10}^{-6},$$

(13)

where ${\sigma }_k$ denotes the molecular collision diameter, and ${\Omega }_{\mu ,k}$ represents the temperature-dependent collision integral. The thermal conductivity of each species is obtained via its corresponding Prandtl number,

$${\kappa }_k={\displaystyle \frac{C_{p,k}}{P{r}_k}}{\mu }_k.$$

(14)

The binary diffusion coefficient for species i and j is evaluated as

$$D_{ij}={\displaystyle \frac{0.0266}{{\Omega }_D}}{\displaystyle \frac{T^{3/2}}{p\sqrt{M_{ij}}{\sigma }_{ij}^2}},$$

(15)

where $M_{ij}={\displaystyle \frac{2M_i{M}_j}{M_i+M_j}}$ and ${\sigma }_{ij}=\frac{1}{2}({\sigma }_i+{\sigma }_j)$.

The species viscosity is evaluated by means of a Chapman–Enskog-type expression [50],

$${\mu }_k=C_{\mu }{\displaystyle \frac{\sqrt{M_{k}T}}{{\sigma }_k^2}}{\mathrm{\Phi }}_{\mu }^{-1}\left(T_k^{★}\right),C_{\mu }=2.6693\times {10}^{-6},$$

(16)

where $T_k^{★}=T/{(ϵ/K_B)}_k$ is the normalized temperature, and ${\mathrm{\Phi }}_{\mu }(·)$ denotes the temperature-dependent collision–integral function

$${\mathrm{\Phi }}_{\mu }\left(T^{★}\right)=A_{\mu }{\left(T^{★}\right)}^{-B_{\mu }}+C_{\mu }^{★}exp\left(-D_{\mu }{T}^{★}\right)+E_{\mu }exp\left(-F_{\mu }{T}^{★}\right),$$

(17)

with $(A_{\mu },B_{\mu },C_{\mu }^{★},D_{\mu },E_{\mu },F_{\mu })=(1.161,0.149,0.525,0.773,2.162,2.438)$. Thermal conductivity for species k is related to the viscosity via

$${\kappa }_k={\displaystyle \frac{C_{p,k}}{P{r}_k}}{\mu }_k.$$

(18)

The mass-diffusion coefficient of a binary mixture is written as

$$D_{ij}=C_D{\displaystyle \frac{T^{3/2}}{p\sqrt{M_{ij}}{\sigma }_{ij}^2}}{\mathrm{\Phi }}_D^{-1}\left(T_{ij}^{★}\right),C_D=0.0266,$$

(19)

where $M_{ij}=2M_i{M}_j/(M_i+M_j)$, ${\sigma }_{ij}=\frac{1}{2}({\sigma }_i+{\sigma }_j)$, and $T_{ij}^{★}=\sqrt{({ϵ}_i/K_B)({ϵ}_j/K_B)}$. The function ${\mathrm{\Phi }}_D$ represents the diffusion collision integral and is approximated by [47]

$${\mathrm{\Phi }}_D\left(T^{★}\right)=A_D{\left(T^{★}\right)}^{-B_D}+C_D^{★}exp\left(-D_D{T}^{★}\right)+E_{D}exp\left(-F_D{T}^{★}\right)+G_{D}exp\left(-H_D{T}^{★}\right),$$

(20)

with the parameter values

$$(A_D,B_D,C_D^{★},D_D,E_D,F_D,G_D,H_D)=(1.060,0.156,0.193,0.476,1.036,1.530,1.765,3.894)$$

(21)

Although the full viscous NSF equations are employed in the present study, it is important to note that the early-time dynamics of shock-driven interface instability are primarily governed by inviscid baroclinic mechanisms. For the highly transient interactions considered here, the characteristic viscous boundary layer thickness remains small compared to the heavy-layer thickness, and viscous stresses contribute negligibly to the initial vorticity deposition and interface growth. This observation is consistent with earlier studies, which demonstrated that viscous effects play a secondary role during short-time shock–interface interactions and become relevant mainly during later-stage vortex roll-up and dissipation. Nevertheless, viscous terms are retained to provide physical regularization and to accurately capture late-time shear-layer diffusion and enstrophy decay.

2.3. Initial and Boundary Conditions

A planar shock with a prescribed Mach number of $M_s=1.25$ is introduced in the surrounding light gas (N₂) upstream of the heavy-fluid layer. The flow ahead of the shock is kept at uniform reference conditions $(p_0,T_0)$, with $p_0=101,325\mathrm{Pa}$ and $T_0=293\mathrm{K}$. The flow properties across the shock are linked by the classical Rankine–Hugoniot relations, which specify the thermodynamic and kinematic states immediately downstream of the shock.

$$M_2^2={\displaystyle \frac{2+(\gamma -1)M_s^2}{1-\gamma +2\gamma M_s^2}},{\displaystyle \frac{p_2}{p_1}}={\displaystyle \frac{1+\gamma M_s^2}{1+\gamma M_2^2}},{\displaystyle \frac{{\rho }_2}{{\rho }_1}}={\displaystyle \frac{\gamma -1+(\gamma +1)(p_2/p_1)}{\gamma +1+(\gamma -1)(p_2/p_1)}},$$

(22)

where $\gamma =1.4$ corresponds to the ratio of specific heats for nitrogen, while the subscripts 1 and 2 indicate conditions upstream and downstream of the shock, respectively. The resulting post-shock values are imposed at the inflow boundary to sustain a stable planar incident shock as it propagates toward the heavy-fluid layer.

To avoid artificial reflections, a non-reflecting condition is enforced at the downstream boundary. The top and bottom boundaries are treated with zero-gradient conditions so that both acoustic and hydrodynamic disturbances can exit the domain freely. Inflow conditions corresponding to the post-shock state are specified at the left boundary, while the right boundary is configured to permit unobstructed wave outflow. The heavy-fluid interface is initially perturbed using a single-mode sinusoidal profile, and the perturbation amplitude $a_0$ is systematically varied to assess its impact on the interface dynamics. All other parameters, such as shock strength, layer thickness, and fluid properties, remain unchanged to isolate the effects driven solely by the initial amplitude.

2.4. Post-Processing Diagnostics and Integral Measures

To quantitatively analyze the amplitude-driven dynamics of the shocked heavy-fluid layer, a set of post-processing diagnostics is introduced. These measures are derived from the instantaneous flow fields obtained from the numerical simulations and are employed to characterize vorticity generation, circulation dynamics, enstrophy growth, and kinetic energy redistribution. All quantities are evaluated over the two-dimensional computational domain D.

2.4.1. Vorticity Transport Equation

The evolution of vorticity governs the fundamental dynamics of the RMI, influencing interface deformation, mixing, and the transition to nonlinear flow regimes. In the present shocked heavy-fluid layer, the vorticity transport equation (VTE) is derived from the compressible NSF equations to identify the dominant physical processes responsible for vorticity generation and evolution [51], given by

$${\displaystyle \frac{D\mathit{\omega }}{Dt}}=(\mathit{\omega }·\nabla )\mathbf{u}-\mathit{\omega }(\nabla ·\mathbf{u})+{\displaystyle \frac{1}{{\rho }^2}}(\nabla \rho \times \nabla p)+{\displaystyle \frac{\mu }{\rho }}{\nabla }^2\mathit{\omega }.$$

(23)

Here, $\mathit{\omega }=\nabla \times \mathbf{u}$ denotes the vorticity vector, $\mathbf{u}$ is the velocity field, $\rho$ is the fluid density, p represents pressure, and $\mu$ is the dynamic viscosity. The material derivative $D/Dt$ accounts for both local temporal variation and convective transport of vorticity. Each term on the right-hand side corresponds to a distinct physical contribution: the first term describes vortex stretching and tilting due to velocity gradients along the vorticity direction, which vanishes in purely two-dimensional flows but becomes important in three-dimensional settings; the second term reflects compressibility effects through flow divergence; the third term represents baroclinic vorticity generation arising from misaligned pressure and density gradients ($\nabla p\times \nabla \rho \ne 0$), a dominant mechanism during shock–interface interaction that seeds RMI growth; and the final term captures viscous diffusion and production of vorticity, which gain importance in regions with strong shear and sharp gradients near the interface.

2.4.2. Spatially Integrated Vorticity Production Terms

To quantify the relative contributions of different vorticity generation mechanisms, spatially integrated measures of the dominant source terms in the VTE are defined as

$${\omega }_{\mathrm{DIL}}\left(t\right)={\int }_D|-\mathit{\omega }(\nabla ·\mathbf{u})|dxdy,$$

(24)

$${\omega }_{\mathrm{BAR}}\left(t\right)={\int }_D\left|{\displaystyle \frac{1}{{\rho }^2}}(\nabla \rho \times \nabla p)\right|dxdy,$$

(25)

$${\omega }_{\mathrm{VIS}}\left(t\right)={\int }_D\left|{\displaystyle \frac{\mu }{\rho }}{\nabla }^2\mathit{\omega }\right|dxdy,$$

(26)

where ${\omega }_{\mathrm{DIL}}$, ${\omega }_{\mathrm{BAR}}$, and ${\omega }_{\mathrm{VIS}}$ represent the dilatational, baroclinic, and viscous contributions, respectively. These quantities provide global measures of compressibility effects, baroclinic torque, and viscous diffusion during different stages of the instability evolution.

2.4.3. Vorticity Extremes

The instantaneous strength of counter-rotating vortical structures is characterized using the extrema of the scalar vorticity field,

$${\omega }_{max}\left(t\right)=\underset{\mathbf{x}\in D}{max}\omega (\mathbf{x},t),{\omega }_{min}\left(t\right)=\underset{\mathbf{x}\in D}{min}\omega (\mathbf{x},t).$$

(27)

Here, $\omega (\mathbf{x},t)=(\nabla \times \mathbf{u})(\mathbf{x},t)$ denotes the two-dimensional scalar vorticity field, and D refers to the computational domain. Within the numerical discretization, these values are evaluated at every time step as

$${\omega }_{max}\left(t^n\right)=\underset{i,j}{max}{\omega }_{i,j}\left(t^n\right),{\omega }_{min}\left(t^n\right)=\underset{i,j}{min}{\omega }_{i,j}\left(t^n\right).$$

(28)

These extrema provide direct insight into the intensity, asymmetry, and temporal evolution of baroclinically generated vortices along the perturbed interface.

2.4.4. Circulation Measures

Circulation is employed as a global diagnostic for quantifying the net vorticity deposited by shock–interface interaction. The total circulation over the domain is defined as

$${\mathrm{\Gamma }}^{\mathrm{tot}}\left(t\right)={\int }_D\omega dxdy.$$

(29)

To distinguish between clockwise and counter-clockwise rotating structures, the circulation is decomposed into positive and negative components,

$${\mathrm{\Gamma }}^{\mathrm{pos}}\left(t\right)={\int }_{\omega >0}\omega dxdy,{\mathrm{\Gamma }}^{\mathrm{neg}}\left(t\right)={\int }_{\omega <0}\omega dxdy,$$

(30)

such that ${\mathrm{\Gamma }}^{\mathrm{tot}}\left(t\right)={\mathrm{\Gamma }}^{\mathrm{pos}}\left(t\right)+{\mathrm{\Gamma }}^{\mathrm{neg}}\left(t\right)$. These measures enable assessment of circulation balance and asymmetry arising from nonlinear interface deformation.

2.4.5. Enstrophy

The overall intensity of vortical activity is quantified using the enstrophy, defined as

$$\Omega \left(t\right)={\int }_D{\omega }^{2}dxdy.$$

(31)

Enstrophy serves as a global indicator of small-scale vorticity amplification and mixing, particularly during the nonlinear stages of the RMI.

2.4.6. Kinetic Energy

The redistribution of mechanical energy within the flow is characterized by the total kinetic energy,

$$E_k\left(t\right)={\int }_D{\displaystyle \frac{1}{2}}\rho \left(u^2+v^2\right)dxdy,$$

(32)

where $\rho$ is the local fluid density, and $(u,v)$ denote the streamwise and transverse velocity components, respectively. This measure captures the transfer of shock-induced kinetic energy into interfacial deformation, shear layers, and vortical structures.

3. Numerical Methodology and Validations

The accurate simulation of shocked fluid interfaces demands numerical schemes that can resolve sharp discontinuities while preserving small-scale flow features generated during instability growth. In this study, a high-order local discontinuous Galerkin (LDG) framework is employed to capture the nonlinear amplitude-driven evolution of the RMI in a heavy-fluid layer. The solver’s accuracy and robustness are first verified through standard benchmark problems before applying it to the present configuration.

3.1. Numerical Methodology

The governing compressible multicomponent NSF equations are solved using an in-house high-order mixed local discontinuous Galerkin (LDG) solver [52]. The computational domain is discretized into uniform Cartesian elements, and within each element the flow variables are approximated by third-order Legendre polynomial basis functions. The use of a high-order modal representation allows precise resolution of steep gradients and interfacial deformations induced by shock acceleration. Volume and surface fluxes are computed using Gauss–Legendre quadrature [53], while inter-element coupling is achieved through the Harten–Lax–van Leer–Contact (HLLC) Riemann solver [54], which ensures robust shock capturing and accurate density interface preservation. Diffusive and auxiliary fluxes are evaluated following the alternating flux strategy proposed by Cockburn and Shu [55], maintaining consistency between the primary and auxiliary equations. Time advancement is carried out with a third-order strong-stability-preserving Runge–Kutta (SSP-RK3) method [56].

To prevent non-physical oscillations near strong gradients and discontinuities, a Krivodonova-type moment limiter [57,58] is employed and selectively activated using a modified TVB (Total Variation Bounded) sensor. The sensor identifies troubled elements by monitoring the variation of modal coefficients within each element and comparing them against a TVB threshold that scales with both the local mesh size and polynomial degree. When the local variation exceeds this threshold, the element is flagged as troubled and the Krivodonova limiter is applied to suppress spurious oscillations. The modification of the classical TVB sensor consists of adjusting the TVB constant and normalization to improve robustness for shock-dominated compressible flows while avoiding excessive limiting in smooth regions. This selective activation ensures that limiting is applied only in the vicinity of shock fronts and sharp material interfaces, thereby preserving the high-order accuracy of the scheme in smooth flow regions. The global Courant–Friedrichs–Lewy (CFL) number is fixed at 0.1 to ensure stability and temporal accuracy. In summary, this numerical configuration enables accurate capture of baroclinic vorticity generation, interface roll-up, and nonlinear deformation associated with amplitude-driven shock–interface interactions.

3.2. Validation Study

Before applying the solver to amplitude-driven interfacial instability, its accuracy was assessed through a classical benchmark involving a shocked dense gas cylinder. In this setup, a planar shock of Mach number $M_s=1.22$ strikes an ${\mathrm{SF}}_6$ gas cylinder surrounded by nitrogen, corresponding to a benchmark configuration first examined experimentally by Haas and Sturtevant [59] and subsequently simulated numerically by Quirk and Karni [60]. This case serves as a stringent test for evaluating both the shock-capturing capability and the scheme’s fidelity in predicting interface deformation and vorticity generation.

Figure 2 presents the schlieren comparison between experimental observations, reference simulations, and the present LDG results at several time instants (55–247 μs). Initially, the incident shock interacts with the upstream surface of the heavy cylinder, producing transmitted and reflected waves. As the transmitted shock traverses the dense region, refraction occurs and the interface begins to deform. At intermediate times, baroclinic torque induced along the interface leads to the formation of a jet directed downstream. Subsequently, the interface undergoes strong shear and vortex roll-up, forming distinct mushroom-shaped structures. The present LDG solution reproduces these physical phenomena with high fidelity, including jet penetration, shock refraction, and coherent vortex evolution. For quantitative validation, Figure 3 presents a comparison between experimental and numerical results for the trajectories of major flow features, namely the incident shock (IS), reflected shock (RS), transmitted shock (TS), and the upstream (UI) and downstream (DI) interfaces. The predicted front locations and propagation speeds align closely with the measurements, demonstrating that the solver captures both the kinematic and dynamic evolution of the interface during shock interaction with the dense cylinder.

Further validation is provided in Figure 4, which shows the temporal variation in the circulation magnitude compared with the results of Quirk and Karni [60]. The strong correlation between the two datasets confirms the solver’s accuracy in representing baroclinic vorticity generation and its time-dependent growth after shock impact. The close match across schlieren patterns, interface trajectories, and circulation data establishes the reliability of the high-order LDG framework for modeling shock–interface interactions in multicomponent flows.

4. Results and Discussion

4.1. Grid Refinement Study

To verify numerical accuracy and establish grid independence, a systematic resolution study was conducted for the shocked heavy-fluid layer at an amplitude–wavelength ratio of $a_0/\lambda =0.1$. The numerical study employed four levels of uniform Cartesian resolution, namely Grid-A ($200\times 100$), Grid-B ($400\times 200$), Grid-C ($800\times 400$), and Grid-D ($1200\times 600$). Now we define a non-dimensional time variable as

$$\tau ={\displaystyle \frac{t}{t_0}}={\displaystyle \frac{t{V}_i}{\lambda }}.$$

(33)

where $V_i$ is the incident shock velocity, and L is the characteristic wavelength of the interface perturbation.

Figure 5 illustrates the effect of grid refinement on the density field at $\tau =80$. With increasing resolution, the light–heavy interface becomes sharper and the vortical roll-ups along the interface are more distinctly captured. Coarser grids (Grid-A and Grid-B) exhibit excessive numerical diffusion, resulting in smoother interfaces and delayed instability growth. In contrast, Grid-C and Grid-D reveal well-developed Kelvin–Helmholtz instability (KHI) structures and nonlinear interfacial deformation consistent with the RMI. The close agreement between Grid-C and Grid-D results indicates that the solution is effectively grid-independent. Hence, Grid-D ($1200\times 600$) was adopted for subsequent amplitude-driven simulations to ensure both accuracy and computational efficiency.

Figure 6 presents the temporal evolution of the baroclinic vorticity magnitude $\left|{\omega }_{\mathrm{bar}}\right|$ and enstrophy $\Omega$ across the same grid levels. Both diagnostics increase with grid refinement as smaller vortical features and secondary roll-ups are better resolved. On coarse grids, these quantities are underestimated due to numerical dissipation, whereas refinement to Grid-C significantly enhances the predicted values, capturing the transition to the nonlinear regime. Further refinement to Grid-D yields negligible differences, confirming grid convergence for these integral quantities. The $1200\times 600$ grid (Grid-D) was therefore selected for all subsequent simulations to accurately capture amplitude-driven vorticity generation, interfacial roll-up, and mixing dynamics.

Although the density contours in Figure 5 show only subtle visual differences between Grid-C and Grid-D, noticeable discrepancies are observed in the vorticity and enstrophy measures presented in Figure 6. This behavior is expected as density is a relatively smooth primary variable, while vorticity and enstrophy involve first- and second-order spatial derivatives of the velocity field and are therefore more sensitive to grid resolution. Small improvements in resolving velocity gradients near the shocked interface, shear layers, and developing vortical structures can result in amplified differences in vorticity-based quantities, even when density fields appear visually converged. Similar observations have been reported in previous numerical studies [61] of shock-driven interfacial instabilities, where derivative-based diagnostics require finer grids to achieve convergence compared to primary flow variables.

4.2. Flow Field Evolution

Figure 7 presents the flow field visualization of the temporal evolution of the shock–interface interaction for two amplitude–wavelength ratios, $a_0/\lambda =0.025$ and $a_0/\lambda =0.05$, highlighting the amplitude-driven transition mechanisms in the heavy-fluid layer. In both cases, the planar incident shock (IS) propagates from the light-gas region (N₂) into the sinusoidally perturbed heavy layer (SF₆), generating a reflected rarefaction wave (RR) and a transmitted shock (TS₁) upon impact with the upstream interface. As the transmitted shock traverses the dense layer and impinges on the downstream planar interface, a reflected transmitted shock (RTS) and a secondary transmitted wave (TS₂) emerge, establishing a complex multi-wave interaction environment. The coupling of these compressible waves with the perturbed interface produces strong pressure and density gradients that give rise to baroclinic torque, which is responsible for the vorticity deposition initiating the RMI. At the smaller initial amplitude, $a_0/\lambda =0.025$, as shown in Figure 7a, the interface deformation during the early interaction phase ($\tau \le 20$) is modest, and the baroclinic vorticity remains localized near the interface crests and troughs. The induced circulation is relatively weak, leading to nearly symmetric spike–bubble evolution and a delayed onset of secondary shear instabilities. The transmitted shocks retain their planar structure longer, and the interface displacement amplitude grows almost linearly with time, consistent with the linear RMI growth regime. Only at later times ($\tau \ge 50$) do small-scale KH vortices appear along the shear layers, signaling the transition toward weak nonlinearity. In contrast, for the larger-amplitude case, $a_0/\lambda =0.05$, as illustrated in Figure 7b, the interface presents a higher initial slope ($\partial \eta /\partial y$), which enhances the local misalignment between the pressure and density gradients upon shock impact. The enhanced baroclinic mechanism leads to more intense vorticity layers forming along the interface immediately following the passage of TS₁, thereby hastening interface deformation and triggering an earlier transition to nonlinear behavior. The transmitted and reflected waves undergo more significant refraction, causing curvature of the shock fronts and additional compression inside the heavy layer. Consequently, the downstream RTS interacts more intensely with the previously distorted interface, amplifying secondary vortical structures and triggering prominent KH roll-ups. The interface morphology rapidly evolves into large-scale spike and bubble structures accompanied by extensive mixing zones, characteristic of the nonlinear RMI regime.

Figure 8 illustrates the time evolution of shock–interface interaction for the intermediate-amplitude case $a_0/\lambda =0.1$, marking the transition from the weakly nonlinear behavior seen in Figure 7 ($a_0/\lambda =0.025$–$0.05$) to a strongly nonlinear RMI regime. Following the impact of the incident shock on the perturbed interface, the steeper initial geometry intensifies baroclinic torque and vorticity deposition, resulting in much faster interface deformation than in the lower-amplitude configurations. As the transmitted wave exits the downstream interface, an RTS and an internal reflected transmitted shock (IRTS) are generated, intensifying pressure oscillations within the heavy layer—features absent in Figure 7. These repeated reflections promote early KH roll-ups ($\tau \approx 50$) and the formation of large bulges and vortical structures at later times ($\tau \ge 70$), indicating accelerated transition to nonlinear mixing. Thus, the $a_0/\lambda =0.1$ case exhibits stronger baroclinic vorticity, earlier onset of KHI, and more pronounced interface deformation, demonstrating that increasing the initial amplitude amplifies nonlinearity and interfacial mixing intensity.

Figure 9 presents the time evolution of density contours for amplitude–wavelength ratios $a_0/\lambda =0.2$ and $a_0/\lambda =0.4$, illustrating the transition from moderately to strongly nonlinear RMI dynamics. In both cases, the IS interacts with the perturbed heavy layer, generating RR and TS₁ that penetrate into the dense SF₆ region. Compared to the $a_0/\lambda =0.1$ case (Figure 8), the larger initial amplitudes lead to a steeper interface and stronger local pressure–density misalignment, significantly enhancing baroclinic vorticity deposition at early times ($\tau \le 20$). For $a_0/\lambda =0.2$ (Figure 9a), the transmitted wave striking the downstream interface produces RTS and IRTS waves, which generate multiple compression–expansion cycles inside the heavy layer. The repeated internal shock reflections strengthen the vorticity field and hasten the flow’s progression into the nonlinear regime. By $\tau =50$, well-defined KHI structures emerge along the shear layers, and, by $\tau =80$, the interface develops into coherent roll-ups and spike–bubble structures, indicating enhanced mixing relative to $a_0/\lambda =0.1$. For the highest-amplitude case ($a_0/\lambda =0.4$) (Figure 9b), the flow becomes fully nonlinear shortly after shock impact. The transmitted and reflected shocks undergo strong curvature and refraction due to the pronounced interface deformation, while repeated shock reflections within the heavy layer sustain persistent internal pressure oscillations. These interactions produce intense vorticity accumulation, early onset of KHI ($\tau \approx 40$), and rapid interface roll-up into large-scale vortical structures. Compared with $a_0/\lambda =0.1$, both the growth rate and morphological complexity are substantially higher, with the $a_0/\lambda =0.4$ case showing dominant spike–bubble asymmetry and turbulent-like mixing at $\tau =80$. Overall, increasing the initial amplitude from $a_0/\lambda =0.1$ to $0.2$ and $0.4$ amplifies baroclinic vorticity generation, accelerates the onset of secondary instabilities, and enhances the nonlinear coupling between reflected and transmitted shocks. The transition from smooth interface deformation to fully rolled-up vortical motion demonstrates that amplitude growth strongly governs the intensity and topology of shock-driven mixing in heavy-fluid layers.

Figure 10 presents the time evolution of the disturbed interface for various amplitude–wavelength ratios, demonstrating that larger initial perturbations progressively shift the flow response from a linear regime to nonlinear dynamics. At low amplitudes ($a_0/\lambda =0.025$ and $0.05$), the interface evolves symmetrically, with weak baroclinic vorticity and gradual spike–bubble growth, remaining in the quasi-linear regime. At the intermediate amplitude ($a_0/\lambda =0.1$), stronger shock interactions amplify the baroclinic torque, producing noticeable asymmetry, enhanced curvature, and the onset of KHI roll-ups along the shear layers. For higher amplitudes ($a_0/\lambda =0.2$ and $0.4$), the dynamics become fully nonlinear, with strong pressure–density misalignment generating intense vorticity and rapid interfacial roll-up. Multiple wave reflections within the layer further reinforce compression and shear, leading to large-scale vortex pairing and extended mixing zones. At $\tau =80$, the $a_0/\lambda =0.4$ case exhibits a highly folded interface with dominant spike penetration, marking the saturated nonlinear mixing stage of the shocked heavy-fluid layer.

4.3. Vorticity Generation Mechanism

Figure 11 schematically illustrates the mechanism of vorticity generation in a shocked heavy-fluid layer, which governs the onset and evolution of the RMI. Before shock impact, the pressure and density gradients ($\nabla p$ and $\nabla \rho$) are aligned across the light–heavy interface, resulting in hydrostatic equilibrium with no vorticity. When the planar shock strikes the perturbed interface, it distorts the local pressure field, causing misalignment between $\nabla p$ and $\nabla \rho$ and producing a non-zero baroclinic torque ($\nabla \rho \times \nabla p$). This torque deposits oppositely signed vorticity near the crests and troughs, forming counter-rotating vortices that initiate interface deformation. The transmitted and reflected shocks further amplify these vortices, driving spike–bubble growth and nonlinear roll-up. In the post-shock stage, the evolving shear between fluids strengthens KHI, leading to vortex pairing, secondary roll-ups, and enhanced small-scale mixing. The strength and orientation of the generated vorticity depend on the local curvature and amplitude of the interface, with larger perturbations producing stronger baroclinic torque and faster transition to nonlinearity.

Figure 12 presents the temporal evolution of vorticity contours for the shocked heavy-fluid layer at different amplitude–wavelength ratios, highlighting the effect of initial interface amplitude on baroclinic vorticity generation and instability growth. In all cases, the incident shock impinges on the perturbed light–heavy interface, inducing a misalignment between the pressure and density gradients ($\nabla p\times \nabla \rho$), which generates vorticity of opposite signs along the crests and troughs of the interface. For small initial amplitudes, $a_0/\lambda =0.025$ and $0.05$ [Figure 12a,b], the induced vorticity remains weak and localized near the interface, resulting in nearly symmetric deformation with limited shear amplification. The flow stays in the linear regime, where vorticity generation is primarily governed by the instantaneous pressure gradient imposed by the incident and reflected shocks. At $a_0/\lambda =0.1$ [Figure 12c], the vorticity magnitude increases significantly, and alternating vortical layers become more pronounced, leading to visible roll-ups and onset of nonlinear evolution. The deposited vorticity amplifies through the interaction of transmitted and reflected shocks, producing asymmetry between the spike and bubble regions. For larger amplitudes, $a_0/\lambda =0.2$ and $0.4$ [Figure 12d,e], the misalignment between $\nabla p$ and $\nabla \rho$ becomes stronger due to steeper interface slopes, generating higher baroclinic torque and rapid vortex growth. The counter-rotating vortices quickly roll up into coherent structures, and secondary KHI emerges along the shear layers. These cases exhibit intense vorticity accumulation at the interface and subsequent spreading into the heavy fluid, marking a fully nonlinear mixing regime. The increasing asymmetry of the vorticity field and enhanced vortex pairing indicate the transition from orderly interface deformation to complex intense mixing dynamics.

4.4. Spatially Integrated Vorticity Generation Mechanisms

Figure 13 illustrates the time histories of the domain-integrated vorticity source terms obtained from the vorticity transport equation, including the dilatational (${\omega }_{\mathrm{DIL}}$), baroclinic (${\omega }_{\mathrm{BAR}}$), and viscous (${\omega }_{\mathrm{VIS}}$) components. These metrics quantify the respective roles of compressibility effects, baroclinic torque, and viscous diffusion throughout the shock–interface interaction for different amplitude–wavelength ratios $a_0/\lambda$. In the early stage ($\tau <20$), immediately after the incident shock traverses the interface, the baroclinic contribution ${\omega }_{\mathrm{BAR}}$ increases rapidly as a result of pronounced pressure–density gradient misalignment ($\nabla p\times \nabla \rho$). This process governs the initial vorticity deposition and establishes the dominant circulation structures responsible for the onset of the RMI. The magnitude of ${\omega }_{\mathrm{BAR}}$ increases with the initial amplitude $a_0/\lambda$, indicating that larger perturbations enhance the local curvature of the interface and strengthen the baroclinic torque. The high-amplitude cases ($a_0/\lambda =0.2$–$0.4$) show sustained baroclinic activity over longer timescales due to multiple shock reflections and wave–interface interactions within the heavy-fluid layer. The dilatational contribution ${\omega }_{\mathrm{DIL}}$, shown in Figure 13a, grows more gradually and represents vorticity production associated with compressibility effects, including local expansion and compression of the flow field. While ${\omega }_{\mathrm{DIL}}$ remains smaller in magnitude than ${\omega }_{\mathrm{BAR}}$, it plays a secondary role in redistributing vorticity near the shock fronts and rarefaction regions, particularly during the intermediate stage ($20<\tau <50$). In the higher-amplitude configurations, ${\omega }_{\mathrm{DIL}}$ exhibits enhanced fluctuations, reflecting intensified compressible interactions as the interface deformation becomes nonlinear. The viscous contribution ${\omega }_{\mathrm{VIS}}$ [Figure 13c] increases steadily after $\tau \approx 30$ as fine-scale vortical structures and shear layers develop due to KHI roll-ups along the distorted interface. Although the viscous term is weaker than the baroclinic and dilatational components during the initial stages, it becomes increasingly significant in the later nonlinear phase ($\tau >50$), where it governs the diffusion and dissipation of vorticity at smaller scales. Larger-amplitude cases show delayed but stronger viscous response, consistent with the higher shear and enhanced mixing intensity.

Figure 13 shows that the viscous contribution to the vorticity budget becomes noticeable only at later times and remains at least an order of magnitude smaller than the baroclinic contribution. This behavior highlights the fundamentally different roles of baroclinic torque and viscosity in the present flow configuration. Baroclinic vorticity generation arises from inviscid pressure–density misalignment during shock–interface interaction and therefore dominates the early and intermediate stages of the instability. In contrast, the viscous term does not act as a primary source of vorticity but instead governs the redistribution and dissipation of vorticity once small-scale structures and shear layers have developed. As the flow transitions to a more complex vortical state, viscosity becomes increasingly effective in diffusing fine-scale gradients, leading to enstrophy decay and attenuation of small-scale vortices rather than their direct generation. This interpretation is consistent with the delayed onset and smaller magnitude of the viscous contribution observed in Figure 13.

4.5. Temporal Evolution of Vorticity Extrema

Figure 14 displays the time histories of the maximum and minimum vorticity, ${\omega }_{max}$ and ${\omega }_{min}$, for shock-accelerated single-mode heavy-fluid layers across different amplitude–wavelength ratios. These quantities characterize the circulation strength and asymmetry between counter-rotating vortices generated by baroclinic torque during the shock–interface interaction. Immediately after shock impact ($\tau <20$), both extrema rise sharply due to vorticity deposition at regions where pressure and density gradients are misaligned ($\nabla p\times \nabla \rho \ne 0$). Their magnitudes increase with the initial amplitude $a_0/\lambda$, reflecting stronger curvature and greater baroclinic misalignment. For moderate amplitudes ($a_0/\lambda =0.05$–$0.1$), ${\omega }_{max}$ and ${\omega }_{min}$ grow steadily and enter a quasi-periodic state as reflected and transmitted shocks interact within the layer. At higher amplitudes ($a_0/\lambda =0.2$–$0.4$), rapid saturation and large oscillations occur, indicating early nonlinear roll-up, vortex pairing, and enhanced shear along the spike–bubble boundaries. Beyond $\tau \approx 50$, both extrema gradually decay or stabilize due to viscous diffusion and secondary instabilities. The increasing asymmetry between positive and negative vorticity at larger amplitudes signifies dominant rotation on one side of the interface, intensified mixing, and sustained vortex deformation in the nonlinear regime.

4.6. Circulation Analysis

Figure 15 shows the temporal evolution of spatially integrated circulation for different amplitude–wavelength ratios, decomposed into positive ${\mathrm{\Gamma }}^{\mathrm{pos}}$, negative ${\mathrm{\Gamma }}^{\mathrm{neg}}$, and total ${\mathrm{\Gamma }}^{\mathrm{tot}}$ components. They quantify the strength and balance of counter-rotating vortical structures generated by baroclinic torque ($\nabla \rho \times \nabla p$) during the shock–interface interaction and subsequent nonlinear evolution. In the early stage ($\tau <20$), both ${\mathrm{\Gamma }}^{\mathrm{pos}}$ and ${\mathrm{\Gamma }}^{\mathrm{neg}}$ rise sharply as the incident shock deposits baroclinic vorticity along the perturbed interface. The growth rate increases with amplitude ratio $a_0/\lambda$, indicating stronger misalignment of pressure and density gradients and enhanced vorticity generation in higher-amplitude cases. Within $\tau =20$–50, the circulation continues to increase as transmitted and reflected shocks interact within the heavy-fluid layer, promoting asymmetric vortex formation and roll-ups. The magnitudes of ${\mathrm{\Gamma }}^{\mathrm{pos}}$ and ${\mathrm{\Gamma }}^{\mathrm{neg}}$ remain approximately equal in low-amplitude cases, but a clear imbalance appears at higher $a_0/\lambda$, reflecting stronger nonlinear deformation and directional dominance of vortical motion. At later times ($\tau >50$), both ${\mathrm{\Gamma }}^{\mathrm{pos}}$ and ${\mathrm{\Gamma }}^{\mathrm{neg}}$ tend to saturate, while ${\mathrm{\Gamma }}^{\mathrm{tot}}$ [Figure 15c] oscillates around zero, indicating overall conservation of circulation but local asymmetry between clockwise and counter-clockwise vortices. The fluctuations in ${\mathrm{\Gamma }}^{\mathrm{tot}}$ are more pronounced for larger amplitudes, consistent with intensified KHI activity and vortex pairing.

As shown in Figure 15c, the total circulation ${\mathrm{\Gamma }}^{tot}$ remains close to zero and exhibits small oscillations throughout the evolution. These oscillations do not indicate numerical error or boundary-induced artifacts. Rather, they result from the near-cancellation of the large positive and negative circulation components shown in Figure 15a,b, which are associated with counter-rotating vortical structures generated by baroclinic torque. Since ${\mathrm{\Gamma }}^{tot}={\mathrm{\Gamma }}^{pos}+{\mathrm{\Gamma }}^{neg}$ represents the difference between two quantities of comparable magnitude, small physical fluctuations in the vorticity field—particularly during vortex interaction and roll-up—can produce visible oscillations in the net circulation. Importantly, the magnitude of ${\mathrm{\Gamma }}^{tot}$ remains at least an order of magnitude smaller than the individual contributions, indicating that the overall circulation balance is preserved and that boundary effects are negligible in the present configuration.

4.7. Enstrophy Growth Mechanisms

Figure 16 illustrates the effects of amplitude–wavelength ratio on enstrophy distribution in a shocked heavy-fluid layer at $\tau =50$. Enstrophy quantifies the cumulative strength of vortical activity generated through baroclinic torque during the shock–interface interaction. For small amplitudes ($a_0/\lambda =0.025$–$0.05$), enstrophy remains weak and localized near the interface, indicating limited vorticity generation and predominantly linear interface deformation. At moderate amplitude ($a_0/\lambda =0.1$), the enstrophy field becomes more pronounced, with alternating regions of high intensity along the spike–bubble interface, marking the onset of nonlinear roll-ups and vortex pairing. For larger amplitudes ($a_0/\lambda =0.2$–$0.4$), enstrophy amplifies significantly, forming concentrated high-value regions near the shear layers where KHI develops. These strong localized peaks reflect enhanced small-scale vortical interactions and increased mixing intensity. Overall, increasing $a_0/\lambda$ leads to stronger baroclinic vorticity generation, earlier transition to nonlinearity, and higher enstrophy accumulation, signifying a shift from smooth interface deformation to a highly nonlinear mixing regime in the RMI.

Figure 17 shows the time evolution of spatially integrated enstrophy $\Omega$ for different amplitude–wavelength ratios. Enstrophy quantifies the overall vortical intensity in the flow and reflects the cumulative effects of baroclinic torque, vortex stretching, and viscous dissipation during the RMI evolution. At early times ($\tau <20$), $\Omega$ increases rapidly due to the initial baroclinic vorticity deposition caused by the incident shock. For small amplitudes ($a_0/\lambda =0.025$–$0.05$), the growth remains moderate, indicating limited vortex generation and nearly linear interface response. In contrast, higher amplitudes ($a_0/\lambda \ge 0.1$) exhibit stronger and earlier enstrophy amplification due to intensified pressure–density gradient misalignment and enhanced shear at the interface. Beyond $\tau \approx 30$, secondary shock interactions and KHI roll-ups further increase $\Omega$, particularly for $a_0/\lambda =0.2$–$0.4$, where small-scale vortices form along the interface. At later times ($\tau >60$), the enstrophy either saturates or slightly decays, reflecting viscous diffusion and partial energy redistribution in the mixing layer. Hence, increasing $a_0/\lambda$ accelerates the transition from the linear to nonlinear regime and leads to higher enstrophy magnitudes, consistent with enhanced vorticity production and stronger mixing dynamics in the shocked heavy-fluid layers.

4.8. Kinetic Energy Dynamics

Figure 18 illustrates the effects of amplitude–wavelength ratio on the spatial distribution of kinetic energy in a shocked heavy-fluid layer at $\tau =50$. The kinetic energy field reflects the local intensity of momentum transfer and the redistribution of shock-induced energy within the interface region. For small amplitudes ($a_0/\lambda =0.025$–$0.05$), the kinetic energy remains concentrated near the interface, indicating weak interfacial deformation and limited momentum exchange between the light and heavy fluids. As the initial perturbation amplitude increases ($a_0/\lambda =0.1$), the interface undergoes stronger acceleration, resulting in enhanced energy localization near the spike and bubble tips due to intensified shear and baroclinic vorticity generation. At higher-amplitude ratios ($a_0/\lambda =0.2$–$0.4$), the kinetic energy field becomes highly asymmetric, with distinct high-energy streaks along the shear layers and vortex cores, signifying vigorous nonlinear growth and the onset of mixing. The elevated kinetic energy near these regions corresponds to the conversion of bulk flow energy into small-scale rotational motion, driven by KHI roll-ups and secondary shock interactions.

Figure 19 presents the temporal evolution of spatially integrated kinetic energy $E_k\left(\tau \right)$ for different amplitude–wavelength ratios in the shocked heavy-fluid layer. The total kinetic energy quantifies the global redistribution of momentum and the conversion of shock-induced energy into interfacial deformation and vortical motion during the RMI evolution. Immediately after shock impact ($\tau <10$), $E_k\left(\tau \right)$ increases sharply due to the impulsive acceleration of the interface by the incident shock. For small amplitudes ($a_0/\lambda =0.025$–$0.05$), the growth is nearly linear, corresponding to weak interface perturbations and limited baroclinic energy transfer. As $a_0/\lambda$ increases ($0.1$–$0.2$), the interface deformation intensifies, leading to a more rapid rise in $E_k\left(\tau \right)$ driven by enhanced pressure–density gradient misalignment and stronger shear generation. The largest-amplitude case ($a_0/\lambda =0.4$) shows early saturation, indicating that energy transfer is quickly localized in large vortical structures and subsequently moderated by viscous dissipation and secondary instabilities.

4.9. Effect of Viscosity: Inviscid Versus Viscous Comparison

Figure 20 compares inviscid and viscous simulations in terms of the temporal evolution of spatially integrated baroclinic vorticity ${\omega }_{BAR}$ and enstrophy $\Omega$ for the case $a_0/\lambda =0.1$. In Figure 20a, both inviscid and viscous results exhibit nearly identical behavior during the early and intermediate stages, particularly during the shock–interface interaction and the subsequent rapid rise in baroclinic vorticity around $\tau \approx 35$–45. This close agreement indicates that baroclinic vorticity deposition is governed primarily by inviscid pressure–density misalignment, with viscous effects playing a negligible role during the initial instability growth. As time progresses, small deviations between the two curves become visible, with the inviscid case retaining slightly higher vorticity levels. This difference is attributed to viscous diffusion and dissipation, which gradually attenuate small-scale vorticity in the viscous simulation. A similar trend is observed in Figure 20b, where enstrophy growth in both cases remains nearly identical up to the nonlinear transition stage. At later times, the viscous enstrophy exhibits a modest reduction relative to the inviscid counterpart due to enhanced dissipation of fine-scale vortical structures. In summary, viscosity does not significantly influence the early-time baroclinic vorticity generation or instability growth for the highly transient regime considered, while its effects become noticeable only at later stages through dissipation mechanisms. This observation is consistent with prior studies of short-time RMI and supports the use of inviscid modeling for capturing the dominant baroclinic dynamics in the present configuration.

5. Conclusions and Outlook

This study presents a comprehensive numerical investigation of amplitude-driven dynamics in a shocked heavy-fluid layer using a high-order mixed LDG method. The analysis elucidates the influence of the initial perturbation amplitude on baroclinic vorticity generation, circulation evolution, enstrophy amplification, and kinetic energy redistribution during RMI evolution. The results demonstrate that increasing the initial amplitude–wavelength ratio $(a_0/\lambda )$ substantially enhances baroclinic vorticity deposition and accelerates the transition from the linear to the nonlinear regime.

During the early linear stage of evolution, the disturbance growth rate exhibits only a weak dependence on the initial amplitude and remains primarily governed by the shock strength and wavelength. Variations in $(a_0/\lambda )$ mainly influence the duration of the linear regime, with larger amplitudes triggering an earlier onset of nonlinear behavior rather than significantly altering the linear growth rate itself. Small-amplitude interfaces $(a_0/\lambda \le 0.05)$ therefore display nearly symmetric interface deformation and delayed KHI, whereas larger amplitudes $(a_0/\lambda \ge 0.2)$ promote rapid spike–bubble asymmetry, early vortex roll-up, and intensified mixing.

In contrast, the maximum vorticity and circulation exhibit a clear and monotonic dependence on the initial amplitude. Larger values of $(a_0/\lambda )$ lead to stronger pressure–density misalignment during shock–interface interaction, resulting in enhanced baroclinic torque and increased peak vorticity and circulation. Quantitative diagnostics, including circulation, enstrophy, and kinetic energy, confirm that, while the linear-stage growth rate is only weakly amplitude-dependent, the nonlinear evolution, vorticity intensity, and mixing strength scale positively with the initial perturbation amplitude. The study further highlights how nonlinear interactions involving reflected and transmitted shocks amplify secondary instabilities and facilitate energy transfer toward smaller vortical scales.

Building on the present single-mode analysis, future work will extend this framework to multi-mode perturbed interfaces to investigate mode coupling, phase interactions, and spectral energy transfer in complex RMI-driven mixing layers. Such extensions will enable characterization of the transition from coherent vortex roll-ups to broadband turbulence and provide a more realistic understanding of shock-driven mixing processes that are relevant to applications such as inertial confinement fusion, high-speed combustion, and astrophysical flows.