2. Theoretical formulation

In 2-D flows where both acceleration and density are time-dependent, two inviscid and irrotational semi-infinite fluids, initially separated by a single-mode perturbation, are considered. Based on the Layzer potential-flow model (Goncharov & Li Reference Goncharov and Li2005; Mikaelian Reference Mikaelian2009), the governing equations for this system can be expressed as

(2.1) \begin{align} & \qquad\qquad\qquad\qquad\qquad\quad\nabla ^2\phi _i+\frac {\dot {\rho _i}(t)}{\rho _i(t)}=0,\end{align}

(2.2) \begin{align} & \qquad\qquad\qquad\qquad\qquad\frac {\partial {z}}{\partial {t}}+\frac {\partial {\phi _i}}{\partial {x}}\frac {\partial {z}}{\partial {x}}- \frac {\partial {\phi _i}}{\partial {z}}=0,\end{align}

(2.3) \begin{align}& \sum _{i=1}^{2}(-1)^i\rho _i(t)\left \{g(t)z+\frac {\partial {\phi _i}}{\partial {t}}+\frac {1}{2}\left [\left (\frac {\partial {\phi _i}}{\partial {x}}\right )^2+\left (\frac {\partial {\phi _i}}{\partial {z}}\right )^2\right ]\right \}=f(t).\end{align}

Here, the coordinates $x$ and $z$ are perpendicular and parallel to the time-dependent acceleration $g(t)$ , respectively, $\phi _i$ denotes the velocity potential and $\rho _i(t)$ denotes the fluid density. The subscripts $i$ ( $=$ 1, 2) correspond to the two fluids: for bubbles, $i=$ 1 refers to the heavy fluid, $i=$ 2 to the light fluid and $g(t)\gt 0$ ; for spikes, the situation is reversed, with $i=$ 1 corresponding to the light fluid, $i=$ 2 to the heavy fluid and $g(t)\lt 0$ . This definition can simplify the analysis, enabling a consistent treatment of both bubbles and spikes (Zhang & Guo Reference Zhang and Guo2016), and also results in a signed Atwood number $A$ ( $=(\rho _1-\rho _2)/(\rho _1+\rho _2)$ ), with $A\gt 0$ for bubbles, and $A\lt 0$ for spikes. Following Layzer’s approach, the shape of the finger tip is approximated by a parabola, $z(x,t)=\eta (t)+k\xi (t)x^2$ , where $k$ denotes the wavenumber of the initial interface, and $\eta (t)$ and $\xi (t)$ denote the finger amplitude and the finger-tip curvature, respectively.

In the governing equations, (2.1) is essentially Laplace’s equation for the velocity potential $\phi _i$ , with an additional term $\dot {\rho }_i(t)/\rho _i(t)$ representing a dilatation effect due to time-dependent density $\rho _i(t)$ . Physically, this term acts as an effective body force arising from mass conservation under non-stationary density variations. Equation (2.2) represents the kinematic boundary condition at the interface, which enforces that fluid particles initially on the interface remain there for all time. Equation (2.3) represents the dynamic boundary condition, which imposes pressure continuity across the interface. In (2.3), the term $\rho _i(t)g(t)z$ denotes the potential under time-dependent acceleration, while the remaining terms on the left-hand side correspond to the unsteady Bernoulli contributions for fluid $i$ . The function $f(t)$ , an integration constant from pressure matching, is typically set to $f(t) = 0$ by defining a reference pressure datum.

In Layzer’s potential-flow framework, the potential function is often represented by a Fourier series, $\phi _i=\sum _{j=1}^{\infty }P_i(t)^{(j)}\cos (jkx)e^{(-1)^ijkz}$ . In this series, lower-order terms dominate, while higher-order terms contribute minimally due to their exponential decay with increasing order. Retaining more modes enhances the accuracy of early-time finger behaviour but becomes unsuitable for late-time behaviour because the series diverges at large times (this divergence is especially severe for spikes in high Atwood number cases). In our model, the first term of the Fourier series, denoted as $P_i(t)\cos (kx)e^{(-1)^ikz}$ , is used to characterise the dominant behaviour of the finger. Following Zhang & Guo (Reference Zhang and Guo2016), we approximate the contribution of all remaining terms $\sum _{j=2}^{\infty }P_i(t)^{(j)}\cos (jkx)e^{(-1)^ijkz}$ by $Q_i(t)\cos (c(t)kx)e^{(-1)^ic(t)kz}$ , where $c(t)$ is dependent on the density ratio. Finally, the velocity potentials for compressible fluids can be written as

(2.4) \begin{equation} \phi _i=P_i(t)\cos (kx)e^{(-1)^ikz}+Q_i(t)\cos (c(t)kx)e^{(-1)^ic(t)kz}-\frac {\dot {\rho _i}(t)}{2\rho _i(t)}z^2,\,\,\,i=1,\,2, \end{equation}

where the term, $\dot {\rho _i}(t)z^2/[2\rho _i(t)]$ , denoting a density-related component for compressible fluids, is obtained by the solution of (2.1) expanded up to $x^2$ (Goncharov & Li Reference Goncharov and Li2005); and $P_i(t)$ and $Q_i(t)$ , depending on time $t$ , are also functions of $A$ and $g$ .

Substituting (2.4) into (2.2) and expanding the resulting equations through the order of $x^2$ yields

(2.5) \begin{align} \dot {\eta }(t)-(-1)^{i}k\big [P_i(t)e^{(-1)^ik\eta (t)}+c(t)Q_i(t)e^{(-1)^ic(t)k\eta (t)}\big ]+\frac {\dot {\rho }_i(t)\eta (t)}{\rho _i(t)}=0,\end{align}

(2.6) \begin{align} \begin{aligned}& 2\dot {\xi }(t)-6k^2\xi (t)\big [P_i(t)e^{(-1)^ik\eta (t)}+c(t)^2Q_i(t)e^{(-1)^ic(t)k\eta (t)}\big ]+\\ & (-1)^ik^2\big [P_i(t)e^{(-1)^ik\eta (t)}+c(t)^3Q_i(t)e^{(-1)^ic(t)k\eta (t)}\big ]+\frac {2\dot {\rho }_i(t)\xi (t)}{\rho _i(t)}=0. \end{aligned}\end{align}

By solving $P_i(t),\,Q_i(t)$ from the above equations and substituting the resulting expressions into (2.3), one can obtain an ordinary differential equation for the term proportional to $x^2$

(2.7) \begin{align} F_1\frac {\textrm { d}^2\eta }{\textrm { d}t^2}+F_2\frac {\textrm { d}^2\xi }{\textrm { d}t^2}+F_3 \!\left [\!k^2c^2 \! \left(\! \frac {\textrm { d}\eta }{\textrm { d}t} \right)^2 \! +16 \!\left(\!\frac {\textrm { d}\xi }{\textrm { d}t} \right)^2\!\right ]\!+\!F_4\frac {\textrm { d}\xi }{\textrm { d}t}\frac {\textrm { d}c}{\textrm { d}t}\!+\!F_5\frac {\textrm { d}\xi }{\textrm { d}t}\!+\!F_6\frac {\textrm { d}c}{\textrm { d}t}+F_7=0, \end{align}

where $\textrm { d}\eta /\textrm { d}t$ denotes the finger growth rate, and

(2.8) \begin{align}& F_1=2k\xi (-\rho _1+\rho _2)-\frac {kc\rho _1}{1+c+6\xi }-\frac {kc\rho _2}{1+c-6\xi },\end{align}

(2.9) \begin{align} &\qquad\qquad F_2=\frac {2\rho _1}{1+c+6\xi }+\frac {2\rho _2}{1+c-6\xi },\end{align}

(2.10) \begin{align} & \qquad\qquad F_3=\frac {-\rho _1}{(1+c+6\xi )^2}+\frac {\rho _2}{(1+c-6\xi )^2},\end{align}

(2.11) \begin{align} & \qquad\qquad F_4=\frac {-2\rho _1}{(1+c+6\xi )^2}+\frac {-2\rho _2}{(1+c-6\xi )^2},\\[-6pt] \nonumber \end{align}

(2.12) \begin{align}& F_5=\frac {10kc\dot {\eta }\rho _1+2(1+c-4\xi +5kc\eta )\dot {\rho }_1}{(1+c+6\xi )^2}+\frac {-10kc\dot {\eta }\rho _2+2(1+c+4\xi -5kc\eta )\dot {\rho }_2}{(1+c-6\xi )^2}, \\[-12pt] \nonumber\end{align}

(2.13) \begin{align} & F_6=\frac {(-1-6\xi )k\dot {\eta }\rho _1-(2\xi +k\eta +6k\eta \xi )\dot {\rho }_1}{(1+c+6\xi )^2}+\frac {(-1+6\xi )k\dot {\eta }\rho _2-(2\xi +k\eta -6k\eta \xi )\dot {\rho }_2}{(1+c-6\xi )^2},\end{align}

(2.14) \begin{align} F_7 & =-\frac {kc\dot {\eta }\dot {\rho }_1(2kc\eta +c+2\xi +1)}{(1+c+6\xi )^2}-\frac {\dot {\rho }_1^2(kc\eta -2\xi )(kc\eta -c-8\xi -1)}{\rho _1(1+c+6\xi )^2}-2kg\rho _1\xi \nonumber\\ & \quad +\frac {kc\dot {\eta }\dot {\rho }_2(2kc\eta -c+2\xi -1)}{(1+c-6\xi )^2}+\frac {\dot {\rho }_2^2(kc\eta -2\xi )(kc\eta +c-8\xi +1)}{\rho _2(1+c-6\xi )^2}+2kg\rho _2\xi \nonumber\\ & \quad +\frac {\ddot {\rho }_1(2\xi -kc\eta )}{1+c+6\xi }+\frac {\ddot {\rho }_2(2\xi -kc\eta )}{1+c-6\xi }. \end{align}

There are three unknowns, $c(t),\,\xi (t)$ and $\eta (t)$ , in (2.7). To close the model, two additional relations will be introduced. In the work of Zhang & Guo (Reference Zhang and Guo2016), the term $c(t)$ , introduced as a density-ratio-dependent function, primarily ensures convergence for spike growth while exerting negligible influence on bubble growth. For incompressible fluids, $c(t)=$ $[-6+o(1)]\xi (t)$ , where $o(1)$ denotes higher-order terms that are typically negligible, reducing the approximation to $-6\xi (t)$ (Zhang & Guo (Reference Zhang and Guo2016)). However, this formulation fails to capture spike behaviour in variable-density flows at high Atwood numbers, where density variations significantly alter the interfacial dynamics. Specifically, $c(t)=-6\xi (t)$ markedly underestimates the experimental spike growth due to its omission of compressibility effects. To address this limitation, we proposed a modified form

(2.15) \begin{equation} c(t)=[-6+\alpha ]\xi (t), \end{equation}

where the dimensionless parameter $\alpha$ accounts for density variation effects. Drawing from Goncharov & Li (Reference Goncharov and Li2005), $\alpha$ is formulated using the density-related term $\dot {\rho }_i(t)/\rho _i(t)$ , which is crucial to finger growth. Since the spike dynamics is dominated by the heavy fluid (density $\rho _2$ ), we define $\alpha$ as

(2.16) \begin{equation} \alpha =\frac {-\dot {\rho }_2/\rho _2}{\dot {\rho }_1/\rho _1+\dot {\rho }_2/\rho _2}=\frac {-\dot {\rho }_2\rho _1}{\dot {\rho }_1\rho _2+\dot {\rho }_2\rho _1}, \end{equation}

where $\rho _1$ and $\rho _2$ correspond to the light and heavy fluid densities for spikes, respectively. Although $\alpha$ is small compared with the leading term ( $-6$ ), it critically corrects spike growth predictions in variable-density regimes. Note that (2.16) becomes indeterminate ( $0/0$ ) under three conditions, including $A=1\, (\rho _2=0,\,\dot {\rho }_2=0)$ , $A=-1\,(\rho _1=0,\,\dot {\rho }_1=0)$ and $\dot {\rho }_1=\dot {\rho }_2=0$ . In these cases, $\alpha$ is explicitly defined as $\alpha = 0$ .

In the work of Zhang & Guo (Reference Zhang and Guo2016), the finger-tip curvature $\xi (t)$ was approximated as a constant, specifically equal to the quasi-steady curvature $\xi _{qs}$ , which is expressed as $\xi _{qs}$ = $-(3+A)/(12\sqrt {2(1+A)})$ for both bubbles and spikes under arbitrary density ratios. Although this approximation is reasonable for moderate initial amplitudes, it becomes inadequate for smaller initial amplitudes. For example, in the following experiments, the initial Atwood number is 0.85, resulting in $\xi _{qs}$ = $-$ 0.167. For a single-mode case with an initial amplitude of 6 mm and a wavelength of 120 mm, the initial finger-tip curvature $\xi _0$ (defined as $\xi _0=-k\eta _0/2$ , where $\eta _0$ denotes the initial finger amplitude) is calculated as $-0.157$ , which is close to $\xi _{qs}$ = $-$ 0.167. However, when the initial amplitude is reduced to 2 mm while maintaining the same wavelength (120 mm), $\xi _0$ decreases to $-0.052$ , approximately one third of $\xi _{qs}$ = $-$ 0.167. In this situation, the theoretical finger growth is found to deviate significantly from the experimental results, indicating that the assumption $\xi _0=\xi _{qs}$ is inadequate, particularly for smaller initial amplitudes. Therefore, it is essential to examine the evolution of $\xi (t)$ from its initial value $\xi _0$ to the asymptotic value $\xi _{qs}$ .

To formulate the expression of finger-tip curvature, we define $f(\xi ) = {\rm d}\xi /{\rm d}\eta$ . Following the approach of Zhang & Guo (Reference Zhang and Guo2022), a first-order asymptotic expansion of $f(\xi )$ is performed at $\xi =\xi _{qs}$ . This yields

(2.17) \begin{equation} f(\xi )=f(\xi _{qs})+\beta (\xi -\xi _{qs}). \end{equation}

Here, $\beta =f^\prime (\xi _{qs})$ with $f^\prime = {\rm d}f/{\rm d}\xi$ . Since the finger-tip curvature during the quasi-steady stage is insensitive to amplitude growth (Sohn Reference Sohn2004), i.e. $f(\xi _{qs})=0$ , we have

(2.18) \begin{equation} f(\xi )=\beta (\xi -\xi _{qs}). \end{equation}

Using the two conditions, $\xi _0=-k\eta _0/2$ and $\xi _{qs}$ = $-(3+A)/(12\sqrt {2(1+A)})$ (Zhang & Guo Reference Zhang and Guo2016), the solution to (2.18) can be expressed as

(2.19) \begin{equation} \begin{aligned} \xi (\eta )=\xi _{qs}+\big [\xi _0-\xi _{qs}\big ]e^{\beta (\eta -\eta _0)}, \end{aligned} \end{equation}

where $\beta$ remains unknown.

To determine $\beta$ , we summarise key properties of finger curvature from previous studies:

1. (i) for $A$ = 1, the bubble curvature is given by $\xi = (\xi _0+1/6 )e^{-3k(\eta -\eta _0)}-1/6$ (Zhang Reference Zhang1998; Goncharov Reference Goncharov2002);

2. (ii) the bubble curvature in the quasi-steady stage is insensitive to the density ratio (Sohn Reference Sohn2004; Zhang & Guo Reference Zhang and Guo2022; Liu et al. Reference Liu, Zhang and Xiao2023b );

3. (iii) as $A \rightarrow -1$ , the spike curvature in the quasi-steady stage tends to infinity (Zhang Reference Zhang1998).

Following Zhang & Guo (Reference Zhang and Guo2016, Reference Zhang and Guo2022), $\beta$ can be expressed as a Laurent series in terms of $\sqrt {1+A}$ . Keeping only the three leading terms yields

(2.20) \begin{equation} \beta (A)=\frac {h_{-1}}{\sqrt {1+A}}+h_0+h_1\sqrt {1+A}, \end{equation}

where $h_{-1},h_0,h_1$ are three constant coefficients. For $A=1$ , (2.19) reduces to $\xi = [\xi _0+1/6 ]e^{\beta (\eta -\eta _0)}-1/6$ , which matches the expression for finger curvature given in property (i). By comparing these two expressions, we have

(2.21) \begin{equation} \beta (1)=h_{-1}/\sqrt {2}+h_0+h_1\sqrt {2}=-3k. \end{equation}

Based on property (ii), $\xi _{qs}$ is approximately constant for $A\in (0,1]$ . Since $\beta (A)$ represents the higher-order derivative of $\xi$ , it is assumed that $\beta (A)$ is a constant. To satisfy (2.20), $h_1=h_{-1}=0$ . Substituting these values into (2.21) gives $h_0=-3k$ , namely $\beta (A)$ = $-3k$ .

Then, the expression for the variable finger-tip curvature under time-dependent acceleration and density is given by

(2.22) \begin{equation} \ \xi (\eta )=-\frac {3+A}{12\sqrt {2(1+A)}}+\bigg [-\frac {k\eta _0}{2}+\frac {3+A}{12\sqrt {2(1+A)}}\bigg ]e^{-3k(\eta -\eta _0)}, \end{equation}

which holds for all $A\in (-1,1]$ . Equation (2.22) characterises the dependence of finger curvature on the initial interface conditions (including $A$ , $k$ and $\eta _0$ ), as well as the interrelation between the morphology evolution and growth rate of fingers.

Property (iii) is used to further assess (2.22). As $A \rightarrow -1$ , the first term on the right-hand side of (2.22) tends to $-\infty$ , while the remaining term approaches zero in the quasi-steady stage. Therefore, property (iii) is satisfied.

Through simple mathematical transformations, ${\rm d}\xi /{\rm d}t$ and ${\rm d}^2\xi /{\rm d}t^2$ can be expressed respectively as

(2.23) \begin{equation} \ \frac {\textrm { d}\xi }{\textrm { d}t}=\frac {\textrm { d}\xi }{\textrm { d}\eta }\frac {\textrm { d}\eta }{\textrm { d}t},\,\,\, \frac {\textrm { d}^2\xi }{\textrm { d}t^2}=\frac {\textrm { d}^2\xi }{\textrm { d}\eta ^2}\bigg (\frac {\textrm { d}\eta }{\textrm { d}t}\bigg )^2+\frac {\textrm { d}\xi }{\textrm { d}\eta }\frac {\textrm { d}^2\eta }{\textrm { d}t^2}. \end{equation}

Neglecting the higher-order terms results in $\textrm { d}c/\rm dt=(-6+\alpha )\textrm { d}\xi /\rm dt$ . Substituting the expression of $\textrm { d}c/\rm dt$ and (2.23) into (2.7), the finger growth rate ( $v=\textrm { d}\eta /\textrm { d}t$ ), governed by time-dependent acceleration and density, can be obtained by solving the following expression:

(2.24) \begin{equation} \dot {v}=-\frac {S_2v^2+S_3}{S_1}, \end{equation}

where

(2.25) \begin{align} & \,\,\,\qquad\qquad\qquad\qquad S_1=F_1+F_2\frac {\textrm { d}\xi }{\textrm { d}\eta },\end{align}

(2.26) \begin{align} & S_2=F_2\frac {\textrm { d}^2\xi }{\textrm { d}\eta ^2}+F_3\left [k^2c^2+16 \left(\frac {\textrm { d}\xi }{\textrm { d}\eta } \right)^2\right ]+F_4(-6-\alpha ) \left(\frac {\textrm { d}\xi }{\textrm { d}\eta } \right)^2,\end{align}

(2.27) \begin{align} & \qquad\qquad S_3=\left [F_5+F_6(-6-\alpha )\right ]\frac {\textrm { d}\xi }{\textrm { d}\eta }v+F_7.\end{align}

We comment that the result of (2.24) is valid for arbitrary density ratios. It should be emphasised that the setting of $A \gt 0$ and $g \gt 0$ in (2.24) results in the solution for bubbles, while $A \lt 0$ and $g \lt 0$ correspond to the solution for spikes. The rarefaction-induced acceleration and density, required as inputs for (2.24), were recently derived by Gao et al. (Reference Gao, Guo, Zhai and Luo2024), and their expressions are provided in Appendix A. The density expressions, (A2) and (A3), represent the densities in the vicinity of the interface and incorporate both temporal and spatial (interface position) variations. Therefore, the present model is capable of addressing the spatio-temporal variation of interface density in rarefaction-driven RTI. When $\dot {\rho }_i$ vanishes and both $g$ and $\xi$ are constant, the model (2.24) naturally reduces to the model proposed by Zhang & Guo (Reference Zhang and Guo2016), which applies to constant acceleration and density conditions. Furthermore, when $\dot {\rho }_i=0,\,c=1$ and $g$ is constant, (2.24) recovers to the classical Layzer-type model (Zhang Reference Zhang1998), which is applicable for infinite density ratio extremes ( $A=\pm 1$ ).

To sum up, the contributions of the present model to the theoretical modelling of RTI flows are threefold. First, the effect of time-dependent density on finger evolution, which has seldom been considered in RTI flows, is quantified. Second, the present model can describe the growth behaviours of both bubbles and spikes in rarefaction-driven RTI flows, while the previous model was formulated only for bubbles (Mikaelian Reference Mikaelian2009). Third, the present model provides the expression for the time-dependent curvature of the finger tip as it evolves from its initial value to the quasi-steady value, in contrast to the previous model that assumed a constant finger-tip curvature (Zhang & Guo Reference Zhang and Guo2016). To verify the present model, the experiments and numerical simulations are performed in the following sections.

3. Experimental methods

The experiments of rarefaction-driven RTI are conducted in a horizontal rarefaction tube, as depicted in figure 1( $a$ ). This tube, with an inner cross-sectional area of 6 mm $\times$ 120 mm, contains four main sections: a 1.0 m vacuum section, a 0.35 m transition section, a 0.45 m test section and a 1.0 m reflection section. Both the vacuum and reflection sections are sufficiently long to ensure that the interface evolution during the experimental duration is not affected by the reflected waves from the ends of the tube. Before the experiments, a 3 $\unicode {x03BC}$ m-thick polyethylene terephthalate (PET) diaphragm, secured by a clamp device, is used to separate the vacuum and transition sections. As shown in figure 1( $b$ ), the clamp consists of two parts, with the left part featuring a sharp blade-like edge. Air is then extracted from the vacuum section using a vacuum pump. When the pressure inside the vacuum section reduces to the PET diaphragm’s endurance limit, the sharp edge of the clamp completely cuts off the PET diaphragm, forming rarefaction waves that travel downstream along the transition section. Further details of the rarefaction-tube flow can be found in Appendix B.

Figure 1.

Schematics of the experimental apparatus: the horizontal rarefaction tube ( $a$ ), the clamp device used to secure the diaphragm ( $b$ ) and the interface formation device ( $c$ ). Here, SHO coating denotes the super-hydrophobic-oleophobic coating, used to confine the soap film to the designed shape.

Table 1.

Initial experimental interface parameters. Here, $a_0$ and $\lambda$ represent the initial amplitude and wavelength of the single-mode interface.

In the experiments, to ensure that the evolving interface remains within the rarefaction region, the rarefaction tail must move upstream along the vacuum section after the diaphragm rupture. According to one-dimensional (1-D) gas dynamics theory (Han & Yin Reference Han and Yin1993), when the pressure ratio $p_2/p_1$ (where $p_2$ and $p_1$ denote the pressures in the transition and vacuum sections, respectively) exceeds 5.969, the rarefaction tail propagates upstream under the present experimental conditions (see Appendix B for the determination of $p_2/p_1$ ). In our experiments, the diaphragm ruptures at 10 kPa, yielding $p_2/p_1=$ 10, which satisfies the theoretical criterion for upstream movement of the rarefaction tail.

Five different sets of single-mode interfaces, with their initial amplitudes ( $a_0$ ) and wavelengths ( $\lambda$ ) listed in table 1, are taken into account to study the evolution of rarefaction-driven RTI. For each case, the initial amplitude–wavelength ratio ( $a_0/\lambda$ ) does not exceed 0.05, satisfying the small-perturbation hypothesis. The centre of the initial interface for each case is located at 758.5 mm downstream of the diaphragm. The soap-film technique (Li et al. Reference Li, Cao, Wang, Zhai and Luo2023) is adopted to generate the single-mode interfaces. Transparent acrylic plates are used to manufacture the interface formation device. As depicted in figure 1( $c$ ), this interface device consists of two sub-devices, with their edges pre-carved into sinusoidal shapes. To confine the soap film to the designed shape, a super-hydrophobic-oleophobic coating is applied to the sinusoidal edges of the interface device. Then, by pulling a rectangular brush with moderate soap-film solutions on its borders along the sinusoidal edges, a well-defined soap-film interface is generated.

Combinations of SF $_6$ /air and air/helium gas are commonly used for generating heavy/light interfaces in previous experimental studies (Brouillette & Sturtevant Reference Brouillette and Sturtevant1989; Jourdan & Houas Reference Jourdan and Houas2005; Zhai et al. Reference Zhai, Dong, Si and Luo2016; Morgan et al. Reference Morgan, Cabot, Greenough and Jacobs2018; Guo et al. Reference Guo, Cong, Si and Luo2022a ,Reference Guo, Si, Zhai and Luo b ; Li et al. Reference Li, Chen, Zhai and Luo2024). These two gas combinations yield initial Atwood numbers ( $A_0$ ) of 0.67 and 0.75, respectively, under pure gas conditions. However, in rarefaction-driven flows, the density on both sides of the interface decreases continuously, causing the SF $_6$ /air or air/helium interface to become progressively indistinct in schlieren imaging during late stages (Gao et al. Reference Gao, Guo, Zhai and Luo2024). This poses a significant challenge for experimental measurements. To improve the schlieren imaging in the experiments, a higher initial density ratio is required. In this work, CF $_4$ and helium are selected. In the experiments, after inserting the interface device with a prepared soap film into the test section, CF $_4$ and helium are injected into the test section through small holes at the transition and reflection sections, respectively. The soap film effectively impedes gas exchange on both sides of the interface, creating a discontinuous CF $_4$ /helium gaseous interface. The inflation rate and duration are strictly controlled to ensure consistent gas concentrations in each experimental run. The volume fractions of both CF $_4$ and helium for each run are maintained at approximately 90 $\%$ , resulting in $A_0$ = 0.85.

A schlieren system with a high-speed camera (FASTCAM SA5, Photron Limited) is used to visualise the flow field. The frame rate of the camera is 30 000 frames per second and the exposure time is 1 $\unicode {x03BC}$ s. The spatial resolution is 0.416 mm pixel $^{-1}$ . Further details about the schlieren system can be found in the previous work (Guo et al. Reference Guo, Cong, Si and Luo2022a ). The ambient pressure and temperature are approximately 101.3 kPa and 293.5 K, respectively.