1 Introduction

The interaction of a shock wave with the interface of two fluids having different densities, specific heat ratios, or differences in other physical parameters, involves the Richtmyer–Meshkov instability (RMI), the Kelvin–Helmholtz instability (KHI), and turbulent mixing. At the beginning of interaction, the RMI dominates the flow and produces many small scale vortices deposited on the interface due to the misalignment of pressure and density gradients. These vortices are accumulated and redistributed to form primary Kelvin–Helmholtz (KH) billows. Then, secondary instabilities as well as tertiary instabilities of KHI occur and the primary KH billows begin to merge, which may finally lead to a turbulent mixing transition.

Shock-driven inhomogeneous flow is widely studied due to its importance in inertial confinement fusion [1], supernova explosions [2], and supersonic combustion [3]. First studied by Richtmyer [4] and later confirmed experimentally by Meshov [5], RMI has been extensively studied theoretically as well as experimentally and numerically [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. Many theoretical models for the growth rate and vortex generations have been proposed and validated by experimental and numerical studies [6,7,8,9,10,11,12,13,14,15]. Recently, many numerical studies have been focused on the turbulent mixing of shock-driven inhomogeneous flow, which are well reviewed in Refs. [42, 43]. The turbulent mixing parameters, such as mixing width, molecular mixing fraction and so on, and the turbulent kinetic energy (TKE) spectrum have been widely discussed in numerical simulations [16,17,18,19,20,21]. Moreover, as an alternative way to reveal the development of turbulence, the kinetic energy transfer, such as the scale-to-scale energy transfer or sub-grid stress energy transfer, has received increasing attention in analyzing the developing processes of turbulence flow and turbulent mixing [22,23,24,25,26,27].

The motivation of this paper is to study the mechanisms of the turbulent mixing of a shock-driven triangular fast-slow (Air–\({\mathrm {SF_6}}\)) gases interface by using the finite volume (FV) method based on the high resolution minimum dispersion and controllable dissipation (MDCD) reconstruction and the Harten-Lax-van Leer-contact (HLLC) Riemann solver. The triangular-shaped interface is a new configuration whose vortex dynamics is not well studied. In the present paper, the reconstruction model (R model) proposed by Niederhaus et al. [13] for shock/bubble interaction flow is extended to predict the circulation for such shock triangular gas–cylinder interaction flow. Additionally, the developing processes of turbulent mixing induced by RMI for the present case are characterized by distinct stages of the development of KHI. Several typical stages, such as the formation of primary KH billows, the development of secondary and tertiary instabilities of KHI and the merging process of KH billows, are clearly demonstrated. Both the decoupled range of the length scales and the TKE spectrum are analyzed to show the occurrence of the turbulent mixing transition in the late stage. Furthermore, the variable-density energy transfer and the merging process of vortices are analyzed to reveal the mechanisms of turbulent mixing transition for the studied shock triangular gas–cylinder interaction flow.

This paper is organized as follows. The numerical method and verification are introduced in Sect. 2. The circulation model for shock triangular gas–cylinder interaction is given in Sect. 3. The morphological stages of fluids mixing and the turbulent mixing transition are discussed in Sect. 4. Mechanisms of turbulent mixing transition for the present case are analyzed in Sect. 5 and the conclusions are stated in Sect. 6.

2 Numerical methods and verification

2.1 Governing equations and numerical method

Proposed by Abgrall [44], the governing equations with consistent treatment of the convective terms at a material interface for two compressible fluids can be written as

$$\begin{aligned}&\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {x_j}}}(\rho {u_j}) = 0, \end{aligned}$$
(1)
$$\begin{aligned}&\frac{{\partial (\rho {u_i})}}{{\partial t}} + \frac{\partial }{{\partial {x_j}}}(\rho {u_i}{u_j} + p{\delta _{ij}}) = \frac{{\partial {\tau _{ij}}}}{{\partial {x_j}}}, \end{aligned}$$
(2)
$$\begin{aligned}&\frac{{\partial E}}{{\partial t}} + \frac{\partial }{{\partial {x_j}}}[{u_j}(E + p)] = \frac{\partial }{{\partial {x_j}}}({u_i}{\tau _{ij}}) \nonumber \\&\quad +\, \frac{\partial }{{\partial {x_j}}}\left[ \kappa \frac{{\partial T}}{{\partial {x_j}}} + \sum \limits _{l = 1}^2 {{h_l}\rho \left( {D_l}\frac{{\partial {Y_l}}}{{\partial {x_j}}} - {Y_l}\sum \limits _{m = 1}^2 {{D_m}\frac{{\partial {Y_m}}}{{\partial {x_j}}}} \right) } \right] , \nonumber \\\end{aligned}$$
(3)
$$\begin{aligned}&\frac{{\partial \rho {Y_l}}}{{\partial t}} + \frac{\partial }{{\partial {x_j}}}(\rho {Y_l}{u_j}) = \frac{\partial }{{\partial {x_j}}}\left( \rho {D_l}\frac{{\partial {Y_l}}}{{\partial {x_j}}}\right) , \end{aligned}$$
(4)
$$\begin{aligned}&\frac{\partial }{{\partial t}}\left( \frac{1}{{\gamma - 1}}\right) + {u_j}\frac{\partial }{{\partial {x_j}}}\left( \frac{1}{{\gamma - 1}}\right) = 0, \end{aligned}$$
(5)

where

$$\begin{aligned} \begin{aligned}&E = \frac{p}{{\gamma - 1}} + \frac{1}{2}\rho {{{{\varvec{u}}}}^2}, \\&p = (\gamma - 1)\rho e, \\&{\gamma = \sum \limits _{l = 1}^2 {\frac{{{Y_l}}}{{{M_l}}}{C_{p,l}}} /\sum \limits _{l = 1}^2 {\frac{{{Y_l}}}{{{M_l}}}{C_{V,l}}}.} \end{aligned} \end{aligned}$$
(6)

In Eqs. (1)–(6), \(\rho \) is the total density, \({{\varvec{u}}}\) is the velocity vector, \(\textit{p}\) is the static pressure, \(\textit{T}\) is the temperature, \(\textit{E}\) is the total energy, \(\textit{e}\) is the internal energy, \(\gamma \) is the ratio of specific heats for the fluid mixture and \( {\tau _{ij}} = \mu (\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}) - \frac{2}{3}\mu \frac{{\partial {u_k}}}{{\partial {x_k}}}{\delta _{ij}}\ \) is the viscous stress tensor. Additionally, for a species with subscript \(\textit{l}\) or \(\textit{m}\), \(\textit{Y}\) is the mass fraction, \(\textit{M}\) is the molecular mass, \({C_p} = \frac{{\gamma {R_\mathrm{u}}}}{{\gamma - 1}}\ \) and \( {C_V} = \frac{{{R_\mathrm{u}}}}{{\gamma - 1}}\ \) are the specific heat constants, (\({R_\mathrm{u}}\) is the universal gas constant), and \( h = (e + p)/\rho \ \) is the enthalpy. The molecular transport properties such as the dynamic viscosity coefficient \(\mu \), the conductivity coefficient \(\kappa \), and the diffusivity coefficient D for \({\mathrm {N_2}}\), \({\mathrm {O_2}}\), and \({\mathrm {SF_6}}\) are taken from Tritschler et al. [16] and Shankar and Lele [45]. The mixing rule of the viscosity conductivity coefficients for gas mixtures used in this paper is proposed by Wilke [46], and has been successfully used for RMI by Giordano and Burtschell [47]. In order to ensure that all diffusive mass fluxes effectively sum to zero, the effective binary diffusivities proposed by Ramshaw [48] are used for each species.

It should be noticed that Eq. (5) is used to ensure numerical compatibility and to suppress the possible oscillations of the numerical solutions in the vicinity of material interfaces [49]. It is otherwise superfluous and used only to compute the pressure. The use of Eq. (5) does not affect the conservation of mass, but it does modify the pressure so as to affect the conservation of momentum and energy. To reduce the conservation error, \(\gamma \) is recovered at every time step using the last formulation of Eq. (6) in this paper. Furthermore, Eq. (5) takes effect only in the vicinity of material interfaces, where the spurious oscillations in the numerical solution will have a significant adverse effect in the prediction of turbulence mixing processes.

In order to resolve small scale structures efficiently, a high resolution FV scheme is applied in the numerical simulation. The fourth-order MDCD reconstruction scheme [50] in conjunction with the HLLC Riemann solver [51, 52] is used to integrate the convective terms, and Green’s theorem is used to integrate the viscous fluxes. The solution is updated temporally by using the third order total variation diminishing (TVD) Runge–Kutta method [53].

2.2 Numerical simulation setup and verification

The flow configuration for the present numerical study is proposed experimentally by Luo et al. [54] and Wang et al. [55] it involves a planar shock of Mach \(M_\mathrm{s}=1.18\) that travels from left to right and impinges on an equilateral triangular gas-cylinder filled with a mixture of 55.0% \({\mathrm {SF_6}}\) and 45% air (in mole fractions). The initial pressure and the temperature are \(1.01\times 10^5\)  Pa and 298 K, respectively, for the unshocked test gas. The height of the equilateral triangular gas cylinder is \(L_0=60\,\hbox {mm}\). The length of the computational domain in the span-wise direction is \( L^{'} =140\,\hbox {mm}\), which corresponds to the span-wise length of the experimental test section. In this simulation, the left and right sides are set to characteristic inflow and outflow conditions, respectively. The periodic boundary conditions are used in the span-wise direction. The flow field configuration at the initial stage is shown in Fig. 1.

Fig. 1
figure 1

Initial flow configuration at the initial stage

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Numerical simulations are conducted on two grids, referred to as the coarse grid (CG) and the fine grid (FG). The fine grid doubles the cell numbers of the coarse grid in both stream-wise and span-wise directions, resulting in grid numbers of about 16 million with 2048 cells in the span-wise direction. It should be noticed that in the present paper, we conducted a two-dimensional simulation. It is well known that the typical turbulent flow is three-dimensional in nature. However, due to the short duration of simulation and fast development of fluids mixing, the flow driven by the KHI is mainly two-dimensional. As discussed by Yosuke and Hoshino [56], even in the two-dimensional KH vortex, the difference in density between two media plays a crucial role in fast turbulent mixing and transport. Consequently, two-dimensional simulations are still useful in studying the path leading to the turbulent mixing transition (or the onset of turbulence). In fact, the turbulent mixing transition and turbulent mixing are widely discussed for two-dimensional RMI-induced flows [20, 21, 57,58,59,60]. To further clarify the two-dimensional nature of the present simulation, we emphasize that the transition and turbulence mentioned in the present paper are the processes of the development of the two-dimensional RMI-induced flow to the extent characterized by rich small-scale structures.

The time histories of the displacement (D) of the leftmost point of the triangular gas-cylinder, the width (L) and height (V) of the gas interface for both grids are shown and compared with those of Luo’s experiment in Fig. 2. As depicted in Fig. 2, the results of displacement, width and height agree well between the two grids, which indicates that the grid resolution is sufficient to resolve the overall features of the flow field. The results also agree reasonably well with the experimental results. To further verify the numerical simulation in this paper, the numerical Schlieren images for the fine grid case at \(t=0.305\,\hbox { ms}\) and \(t=0.495\,\hbox { ms}\) are also compared with the experimental ones in Fig. 3. As depicted in Fig. 3, numerical Schlieren images morphologically agree well with those of the experiment after the tortuous gas interface leaves the large triangular bracket used to form the initial gas interface.

Fig. 2
figure 2

Comparison of displacement and characteristic length

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Fig. 3
figure 3

Comparison of Schlieren images of density. a \(t=0.305\,\hbox { ms}\); b \(t=0.495\,\hbox { ms}\)

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3 An analytical model for predicting the circulation

One of the most important features for shock-driven inhomogeneity flow is the vorticity deposition due to the misalignment of the pressure and the density gradients. Driven by the deposited vorticity, the KHI enhances the mixing of fluids and may lead, finally, to mixing transition. As an integral diagnosis of vorticity, the circulation is a key parameter to determine several characteristic scales that affect the occurrence of turbulent mixing transition (see Sect. 4) [57, 58, 61, 62]. The circulation is defined as

$$\begin{aligned} {\varGamma } = \oint _{{{ P}}} {{{ u}}} \cdot \text {d}{{{ P}}},\ \end{aligned}$$
(7)

which, by the Stokes theorem, is equivalent to the area integral of vorticities given by

$$\begin{aligned} {\varGamma } = \int \int _{{ S}}\,{{\omega }} \, \text {d}{} { x} \text {d}{} { y}. \end{aligned}$$
(8)

\( {\textit{P}} \) is a long enough path enclosing all waves in the upper half-plane and \( {\textit{S}} \) is the area bounded by \( {\textit{P}} \), both of which are shown in Fig. 4. For two-dimensional numerical simulation, the vorticity \( \varvec{\omega } \) is defined as

$$\begin{aligned} \varvec{\omega } = \frac{{\partial u}}{{\partial y}} - \frac{{\partial v}}{{\partial x}}.\ \end{aligned}$$
(9)

The evaluations of numerical circulation over time for the coarse grid and the fine grid, based on the formulation of Eq. (8), are shown in Fig. 5.

Fig. 4
figure 4

Integral path and area for circulation

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As an alternative to numerical simulation, theoretical models based on vortex dynamics are widely used to predict circulation. In the past three decades, many theoretical models, such as the Picone–Boris (PB) model [10], the Yang–Kubota–Zukoski (YKZ) model [11], the Samtaney–Zabusky (SZ) model [12], and R model [13], have been proposed for predicting the circulation of shock-driven inhomogeneity flow. Although these models are derived from inviscid flow, some of them are quite accurate since the viscous effects can be neglected at an early stage [63]. However, most of these theoretical models are formulated for shock-bubble or shock-cylinder interaction. Therefore, some modifications should be made to extend these models for predicting the circulation for the shock triangular gas–cylinder interaction flow. Among all the theoretical models mentioned above, the R model, to some extent, is the most flexible one and can be easily extended since its formulation is based on the theory of one-dimensional (1D) gas dynamics and the exact definition of circulation in Eq. (7).

Fig. 5
figure 5

Evaluation of the numerical results of circulation

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Fig. 6
figure 6

Wave patterns at \(t=0.125\,\hbox { ms}\)

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As analyzed by Luo et al. [54], when the incident shock propagates along the inclined interface of a triangular gas-cylinder as in the present case, the wave patterns of regular refraction with a reflected shock (RRR) can be observed, which is demonstrated by the numerical Schlieren image at time \(t=0.125\,\hbox { ms}\) in Fig. 6. The wave patterns of RRR are characterized by a well-defined refraction point on which the incident shock wave (ISW), the planar transmitted shockwave (TSW), and the reflected shock wave (RSW) are focused. The interaction of the two planar transmitted shock waves results in a Mach stem (MS) and two interaction-induced shock waves (IISW) inside the volume. Based on this analysis of shock wave patterns, a typical schematic diagram is shown in Fig. 7 to extend the R model for computing the circulation of shock triangular gas–cylinder interaction flow. Based on the exact definition in Eq. (7), the circulation can be computed by integrating the velocity along the path A-N-M-F-A, which is large enough to enclose all the waves. The gases along \(\textit{CN, NM}\), and \(\textit{MD}\) are not disturbed by any waves, therefore, the velocities along these lines remain zero. Additionally, the flow along \(\textit{FA}\) is only swept by the initial planar shock wave. Hence, the integration of velocity is zero, since the velocity vector is perpendicular to \(\textit{FA}\). Following the above simplified procedure proposed by Niederhaus et al. [13], the circulation can be computed exactly by using

$$\begin{aligned} {{\varGamma }} = \int _{{A}}^{{B}} {{u_{{\mathrm {AB}} }}} \text {d}x + \int _{{B}}^{{C}} {u_2^{'}}\text {d}x - u_1^{'}({L_3} + {L_4}),\ \end{aligned}$$
(10)

where A, B, and C are, respectively, the positions of RSW, gas interface, and MS on the central line, \( L_3 \) is the stream-wise distance between position A and the leftmost position of the initial interface of the triangular gas-cylinder, and \( L_4 \) is the stream-wise distance between the leftmost position of the initial interface and the ISW. The integral coordinates as well as the control positions and distances are shown in Fig. 7. Additionally, \(u_1^{'}\) is the velocity of shock air behind the ISW, \(u_2^{'}\) is the uniform velocity of shocked \(\mathrm{SF_6}\) between B and C (neglecting the effects of shock refraction and focusing on the central line), and \( u_{\mathrm{AB}} \) is the velocity along \({\textit{AB}}\).

Fig. 7
figure 7

Schematic diagram for computing circulation

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For the shock triangular gas–cylinder interaction flow, we assume that its reflected shock wave can be treated approximately as a cylindrical expanding wave. Consequently, we follow the procedure of Niederhaus et al. [13] to model the velocity along \({\textit{AB}}\) nonlinearly to account for the cylinder expanding effect by fitting a quadratic function of x as

$$\begin{aligned} u_{\mathrm {AB}}(x)=u_1^{'} - \left( u_1^{'} - u_2^{'}\right) {\left( \frac{x}{{2{L_0}}}\right) ^{2}},\ \end{aligned}$$
(11)

where \( L_0 \) is the initial length in the stream-wise direction of the triangular interface, serving as the scaling factor for the nonlinear term of Eq. (11) similar to the radius scaling factor for shock bubble/cylinder interaction flow proposed by Niederhaus et al. [13].

Substituting the modeled \(u_{\mathrm {AB}} \) into Eq. (10) and evaluating the remaining line integrals, one can obtain a circulation formulation of the shock triangular gas–cylinder interaction flow, which is expressed as

$$\begin{aligned} {{\varGamma }} = u_1^{'}{L_1} - \frac{2}{3}{L_0}\left( u_1^{'} - u_2^{'}\right) {\left( \frac{{{L_1}}}{{2{L_0}}}\right) ^3} + u_2^{'}{L_2} - u_1^{'}({L_3} + {L_4}).\ \end{aligned}$$
(12)

After computing the reflected shock wave strength \( M_r \) and speed \( W_r \), the velocity of the Mach stem \( W_t \) and \( u_2{'} \) by solving the 1D gas dynamics equations iteratively for the transmission of a normal shock wave into a slab of gas whose properties are known, the lengths of the line segments \( L_1 \), \( L_2 \), \( L_3\), and \( L_4 \) are calculated using [13]

Fig. 8
figure 8

The numerical Schlieren image at \(t=0.247\,\hbox {ms}\)

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$$\begin{aligned} \begin{aligned}&{L_1} = \frac{1}{{\sqrt{{M_r}} }}\left( u_2^{'} + {W_r}\right) t,\\&{L_2} = \left( {W_t} - u_2^{'}\right) t,\\&{L_3} = \frac{{{W_t}}}{{\sqrt{{M_r}} }}t,\\&{L_4} = {W_i}t. \end{aligned} \end{aligned}$$
(13)
Table 1 Comparison of circulation magnitude
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Fig. 9
figure 9

Vorticity contours at the first stage. a \(t=0.225\,\hbox { ms}\); b \(t=0.275\,\hbox { ms}\)

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where t is the interaction time and \( W_i \) is the velocity of the incident shock wave.

The total interaction time \(t^{\mathrm {*}}\) is defined as the duration between the time at which the ISW begins to interact with the triangular gas-cylinder and the time at which the diffraction shock wave passes the centerline of the vertical edge of the triangle. It is in this duration that Eq. (12) is valid. As discussed by Samtaney and Zabusky [12] and Niederhaus et al. [13], a moderately accurate formulation for the total interaction time \(t^{\mathrm {*}}\) is important for calculating the circulation induced directly by shock–interface interaction. For the flow of shock triangular gas–cylinder interaction with RRR wave patterns, the interaction time before the ISW reaches the vertical edge of the triangle can be directly computed by

$$\begin{aligned} {t_1} = \frac{{{L_0}}}{{{W_i}}}. \end{aligned}$$
(14)

After this moment, a diffraction shock will be formed between the ISW and the vertical edge. Therefore, we need to compute the duration of shock diffraction in order to obtain the total interaction time. Inspired by the work of Whitham [64], we remark that the duration of the shock diffraction process can be predicted by

$$\begin{aligned} {t_2} = \frac{{{L_d}}}{{{M_w}{a_1}}}, \end{aligned}$$
(15)
Fig. 10
figure 10

Contours of air density at the second stage of fluids mixing. a \(t=0.275\,\hbox { ms}\); b \(t=0.300\,\hbox { ms}\); c \(t=0.350\,\hbox { ms}\)

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Fig. 11
figure 11

Contours of air density at the third stage of fluids mixing. a \(t=0.500\,\hbox { ms}\); b \(t=0.650\,\hbox { ms}\); c \(t=1.000\,\hbox { ms}\); d \(t=1.500\,\hbox { ms}\)

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Fig. 12
figure 12

Time history of outer-scale Re

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Fig. 13
figure 13

Plot of lgRe versus lg(\({\lambda }/{\delta }\))

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where \( M_w \) is the so called “wall Mach number”, \(a_1\) is the sound speed of ambient air and \(L_d\) is the half-length of the bottom of the triangular gas-cylinder given by

$$\begin{aligned} {L_d} = {L_0}\tan (\varphi /2),\ \end{aligned}$$
(16)

where \( \varphi = \uppi /3\) is the vertex angle of the triangular gas-cylinder.

As discussed by Whitham, the angle of diffraction and the Mach number of incident shock \( M_\mathrm{s}\) are the two main factors affecting the “wall Mach number”. For a small angle of diffraction, the “wall Mach number” can be estimated using

$$\begin{aligned} {M_w} = \left\{ \begin{array}{lll} {M_\mathrm{s}}+{\theta _w}\left[ \frac{1}{2}(M_\mathrm{s} - 1)\right] ^{2}, &{}\quad \mathrm{if} &{}\quad {{M_\mathrm{s}} \rightarrow {1^ + ,}} \\ {M_\mathrm{s}}+0.4439{M_\mathrm{s}}{\theta _w}, &{}\quad \mathrm{if} &{}\quad {{M_\mathrm{s}}~{>}\!{>}~ 1.} \end{array} \right. \end{aligned}$$
(17)

For a moderate large angle of diffraction, the “wall Mach number” is given as

$$\begin{aligned} {M_w} = \left\{ \begin{array}{lll} 1, &{}\quad \mathrm{if} &{}\quad M_\mathrm{s}\leqslant \exp \left( \frac{{ - {\theta _w}}}{{\sqrt{5.074} 3}}\right) , \\ {M_\mathrm{s}}\exp \left( \frac{{{\theta _w}}}{{\sqrt{5.0743} }}\right) , &{}\quad \mathrm{if} &{}\quad M_\mathrm{s}> \exp \left( \frac{{ - {\theta _w}}}{{\sqrt{5.074} 3}}\right) . \end{array} \right. \end{aligned}$$
(18)

As to the weak shock triangular gas–cylinder interaction with a large diffraction angle \( \theta _w \approx -\,\uppi /2\) as in the present case, the corresponding “wall Mach number” is unity. Therefore, Eq. (15) can be calculated by

$$\begin{aligned} {t_2} = \frac{{{L_0}\tan (\varphi /2)}}{{{a_1}}}. \end{aligned}$$
(19)

Then, the total interaction time is given by

$$\begin{aligned} {t^{\mathrm {*}}} = {t_1} + {t_2} = \frac{{{L_0}}}{{{W_i}}} + \frac{{{L_0}\tan (\varphi /2)}}{{{a_1}}}, \end{aligned}$$
(20)

which is \( t^{\mathrm {*}} =0.247\,\hbox { ms}\) for the present case. Figure 8 shows the numerical Schlieren image at \( t =0.247\,\hbox { ms}\), which indicates that Eq. (20) is quite accurate for predicting the total interaction time. Table 1 lists the magnitude of circulation obtained by the numerical simulation and the present model at three time steps (\( t =0.125\), 0.150, and 0.247 ms). As shown in Table 1, the integrated circulation of the present model agrees well with the simulation result (fine grid), not only at the time before the inclined shock wave begins to diffract (\( t=0.125, 0.150\,\hbox { ms}\)) but also at the final time (\( t =0.247\,\hbox { ms}\)) of the shock–interface interaction. Additionally, the results of the present model become more accurate at the later stage of the duration, which implies that Eq. (11) becomes more efficient to account for the nonlinear effect of expanding RSW. Moreover, as reported by Niederhaus et al. [13], the vorticities are mainly generated during the shock–interface interaction. This also can be observed in the numerical results shown in Fig. 5, in which the variation of the circulation is relatively small after \( t =0.247\,\hbox { ms}\). Consequently, the circulation at \( t =0.247\,\hbox { ms}\) given by present model can be potentially used as the reference value in the criterion of turbulent mixing transition (see Sect. 4).

4 Development of interfacial instability

4.1 Interfacial instability

The development of interfacial instability can be identified by several distinct stages. In the first stage, KHI is induced by RMI. During the interaction of the shock wave with the gas interface, RMI dominates the flow and a large number of small scale baroclinic vorticities are deposited on the gas interface [63]. Then, these vorticities are accumulated and redistributed to form the primary KH billows, which is shown by the evolution of vorticities in Fig. 9. At \( t =0.225\,\hbox { ms}\) (Fig. 9a), there are many small scale vorticities deposited on the gas interface as the ISW passes through the interface. However, at \( t =0.275\,\hbox { ms}\) (Fig. 9b), these small scale vorticities are accumulated and redistributed into the core of the primary vortex. This stage can be regarded as the very beginning of KHI [65, 66].

The second stage is characterized by the development of KHI. During this stage, the fluids from both the heavy and light gases are entrained into the core of KH vortices causing the amplitude growth of the primary KH billows, as depicted in Fig. 10a, b. Following the process of amplitude growth, several secondary instabilities, such as the secondary convective instability (SCI) in over-turned regions of the billow cores and the secondary shear instability (SSI) in the braids of the billow [56, 67, 68], start growing at \( t =0.350\,\hbox { ms}\), as depicted in Fig. 10c.

The third stage is the development of the interfacial turbulence structure. As shown in Fig. 11a, b, after \(t=0.5\,\hbox { ms}\), more complex tertiary instabilities occur and the primary KH billows begin to break down and merge with each other. The merging process of KH billows enhances the mixing of the fluids and would lead to turbulent mixing transition, which is well shown in Fig. 11c, d.

Fig. 14
figure 14

Turbulent kinetic energy spectrum

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Fig. 15
figure 15

Three nonlinear transfer components in Fourier space. a Quadratic component; b pressure component; c dilatational component

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Fig. 16
figure 16

Three nonlinear transfer components in physical space. a Quadratic component; b pressure component; c dilatational component

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Fig. 17
figure 17

Distributions of integral spectrum of quadratic, pressure components and their sum along the stream wise direction. a Integrated quadratic term and pressure term; b combined effect; c pressure contour

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4.2 Mixing transition

To verify the occurrence of the mixing transition, we use the theory of Zhou et al. [58] and Dimotakis [61]. According to this theory, the “mixing transition” coincides with the appearance of an inertial range of scale, which requires the Liepmann–Taylor scale \(\lambda _L\) being larger than the inner-viscous scale \( \lambda _v \) [61], and consequently, there is a decoupling between these scales. These two scales are defined as

$$\begin{aligned} {\lambda _L}= & {} 5.0R{e^{ - 1/2}}\delta , \end{aligned}$$
(21)
$$\begin{aligned} {\lambda _v}= & {} 50R{e^{ - 3/4}}\delta , \end{aligned}$$
(22)

where \( \delta \) is the largest scale in the flow and Re is the outer-scale Reynolds number. Based on circulation, the outer-scale Re is given by [58]

$$\begin{aligned} Re = \frac{{\left| \varGamma \right| }}{\upsilon },\ \end{aligned}$$
(23)

where \(\varGamma \) is the circulation in the half dominant in the span-wise direction and \(\upsilon \) is the kinematic viscosity [59]. For unsteady flow, the outer-scale Re is time-dependent and its time history is shown in Fig. 12. It should be noted that, although the circulation is obtained here from numerical simulation, the analytical model discussed in Sect. 3 provides an alternative way to calculate the circulation approximately up to \(t=t^*\). The length scales also change over time. A plot of lgRe versus lg(\({\lambda }/{\delta }\)) is shown in Fig. 13. At the earlier stage, the outer-scale Re is lower than the threshold Re of \(10^4\) proposed by Dimotakis [61]. Consequently, the regions corresponding to the Lipeman–Taylor scale and the inner viscous scale overlap, which implies that there is no scale separation. As the outer scale Re increases, an uncoupled range appears and is extended. Hence, the inertial range is broadened and the so called “turbulent mixing transition” for two-dimensional RMI-induced flow, which is initially discussed by Zhou et al. [57], is triggered.

The broadened inertial range with increasing time can also be analyzed in Fourier space by investigating the stream-wise averaged turbulent kinetic energy spectra \( { TKE} (k) = \frac{1}{L}\int _\mathrm{IMZ} {\frac{1}{2}{\underline{{u_i}^{'}}} \, {{\underline{{u_i}^{'}} }^*}} \text {d}x \). In the above definition, \( \underline{u_i^{'}} \) is the Fourier transform of \( u_i^{'} = {u_i} - {<}\rho {u_i}{>}_y{/}{<}\rho {>} _y \), \( {\underline{u_i^{'}} ^*} \) is the corresponding complex conjugate of transform of \( \underline{u_i^{'}} \), and \( L = \int _\mathrm{IMZ} {} \text {d}x \) is the length of the inner mixing zone (IMZ) which is bounded by the region with \( {\mathrm {SF}}_6 \) mass fraction \( {Y_{{\mathrm {SF}}_6}} \in [0.05,0.95] \). For scalar \( \phi \), \( {<}\phi {>} _y \) is the ensemble average in the span-wise direction in IMZ. The decaying law of stream-wise averaged turbulent kinetic energy spectra is widely discussed to demonstrate the interfacial turbulent mixing zone of two-dimensional RMI-induced flow [20, 21, 59, 60]. However, based on the above published reports, it would vary from \(k^{-5/3}\) to \(k^{-3}\) as the wave number increases, if the two-dimensional turbulent mixing zone is formed. The stream-wise averaged turbulent kinetic energy spectra at four different time steps for the present simulation is shown in Fig. 14. At an early time (\(t=0.3\,\hbox { ms}\)), the decaying law of \(k^{-5/3}\) for TKE is obscure. However, the TKE decays with a slope of \(k^{-5/3}\) in a broader range of low wavenumbers at a later time (\( t =1.5\,\hbox { ms}\)). Moreover, the decaying law of \(k^{-3}\) for TKE at high wavenumbers reported by Thornber and Zhou [20], Olson and Greenough [21], and Claude and Serge [60] for two-dimensional shock-driven turbulent mixing is also observed at the later time in the present simulation. This implies that the inertial range is extended during the developing process and manifests the appearance of turbulent mixing at later times for the present two-dimensional RMI-induced flow.

5 Mechanisms of turbulent mixing transition

The development of turbulent mixing transition is closely related to the second and third stages described in the previous section, namely the growth of the KH billows and the merging process of the KH vortices. In this section, these two processes are further studied to reveal the main mechanism leading to the mixing transition.

5.1 Energy transport

To study the mechanism of the second stage, we consider the budget of the energy transport equation. For compressible flow, the transport equation of variable-density kinetic energy in Fourier space is proposed by Zhou et al. [23, 24]. The energy budget equation at a fixed stream-wise location for two-dimensional RMI flow yields

$$\begin{aligned} \frac{\partial }{{\partial t}}E(k,x,t) = T(k,x,t) + \varepsilon (k,x,t),\ \end{aligned}$$
(24)

where E, T, and \( \varepsilon \) are, respectively, the variable-density kinetic energy spectrum, the nonlinear transfer term, and the viscous dissipation term for span-wise wave number k at the stream wise location x. As addressed by Thornber and Zhou [24], nonlinear transfer dominates the energy transfer process and the viscous dissipation term is almost negligible for the energy variable-density kinetic budget. Therefore, the nonlinear transfer term is investigated in this paper. In the above equation, the variable-density kinetic energy reads

$$\begin{aligned} E = \frac{1}{2}\underline{{v_i}}\,{\underline{{v_i}} ^{*}},\ \end{aligned}$$
(25)

where \( v_i=\rho ^{1/2}{u_i} \). The nonlinear transfer term is given by

$$\begin{aligned} T = {T_q}(k,x,t) + {T_p}(k,x,t) + {T_d}(k,x,t),\ \end{aligned}$$
(26)

where \( T_m={\underline{{v_i}} ^{*}}\,{\underline{{N_i}^{m}} } + {\underline{{v_i}}}\,{\underline{{N_i}^{m}} }^{*} \). The subscript \(m=q, p, d\) stands for quadratic (convective), pressure, and dilatational components of the nonlinear transfer term, respectively. And \(N_i^m\) reads

$$\begin{aligned} N_i^q = - \frac{{\partial {v_i}{u_j}}}{{\partial {x_j}}},\ N_i^p = - {\rho ^{ - 1/2}}\frac{{\partial p}}{{\partial {x_i}}},\ N_i^d = \frac{1}{2}{v_i}\frac{{\partial {u_k}}}{{\partial {x_k}}}.\ \end{aligned}$$
(27)

In the above definitions, \((\underline{\cdot })\) indicates the Fourier transform of \((\cdot )\) and \((\underline{\cdot })^*\) indicates the complex conjugate of the transform of \((\underline{\cdot })\). Additionally, the required derivatives are computed using second-order central differences.

Figure 15a–c shows the quadratic, pressure, and dilatational components, respectively, of the nonlinear transfer term for the transport equation of variable-density kinetic energy at \(t=0.300\,\hbox { ms}\) in Fourier space. As shown in Fig. 15, due to the weak compressibility of the flow, the dilatation term is much smaller than the quadratic and pressure terms in Fourier space (see the contour level). Additionally, the quadratic and pressure spectrums are distributed visibly with alternating positive and negative regions along the stream-wise direction. The above two characteristics are similar to those discussed by Thornber in three-dimensional RMI flow [24]. To see these regions more clearly, we derive the quadratic, pressure, and dilatational terms for the kinetic energy transfer equation in physical space, which are given by

$$\begin{aligned} {T_{q,p}} = -\, {v_i}\frac{{\partial ({v_i}{u_j})}}{{\partial {x_j}}},\ {T_{p,p}} = -\, {u_i}\frac{{\partial p}}{{\partial {x_i}}},\ {T_{d,p}} = \frac{{\rho u_i^2}}{2}\frac{{\partial {u_k}}}{{\partial {x_k}}}.\ \end{aligned}$$
(28)

The above three terms also can be computed by integrating the corresponding terms in the Fourier space over the entire wavenumber. Figure 16a–c depicts these terms in physical space. It can be shown clearly that the energy transfer mainly happens inside the KH vortices and depends on the former two terms (also see the contour level). To be more specific, to the upstream of a vortex core, the quadratic term tends to be negative and the pressure term positive, while to the downstream of a vortex core, a reverse tendency can be observed. In other words, the convection effect corresponding to the quadratic terms is counteracted by the pressure effect. Therefore, its necessary to study the combined effect of the quadratic and the pressure components, since the total energy transfer rate is mainly determined by this combined effect. Furthermore, the high wavenumber structures are generated through the nonlinear interactions between structures with lower wavenumbers, and the energy is ultimately extracted from the mean flow. Taking these into consideration, we study this combined effect by neglecting the contribution of the mean flow, which is achieved by calculating the integrated quadratic and pressure terms using

$$\begin{aligned} {T_{q,k}}(x) = \sum \limits _{{k^{'}} = 1} {{T_q}({k^{'}},x)}, \end{aligned}$$
(29)
$$\begin{aligned} {T_{p,k}}(x) = \sum \limits _{{k^{'}} = 1} {{T_p}({k^{'}},x)}, \ \end{aligned}$$
(30)
Fig. 18
figure 18

Typical vortex trajectories

Full size image
Fig. 19
figure 19

Density contour and corresponding TKE spectrum contour at different instants. a \(t=0.500\,\hbox { ms}\); b \(t=0.650\,\hbox { ms}\); c \(t=1.000\,\hbox { ms}\); d \(t=1.500\,\hbox { ms}\)

Full size image

where \( k^{'}=kL^{'}/(2\uppi ) \) is the nondimensional wavenumber. It should be noted that the mean flow term, which corresponds to \(k^{'}=0\), is not included in the above two equations. The distributions of integrated quadratic and pressure components are shown in Fig. 17a. The cancellation effect can again be observed between them. However, referring to Fig. 17c where the locations of the vortices are shown, we find that the combined effect \( T_{qp,k}( x )=T_{q,k}(x)+T_{p,k}(x)\) depicted in Fig. 17b is positive in nearly every vortex and some of the values are very large. This indicates there is energy transported from the mean flow to the smaller scale flow of high wavenumbers inside theses vortices. Furthermore, this energy will be further transported to higher wavenumbers inside the vortices due to the nonlinear effect. This is evident according to the corresponding density contours of \(\mathrm{SF_6}\) and TKE spectrum shown in Fig. 19a where the flow structures with high wave numbers are developed only in the vortex cores.

The above mechanism of the development of the KHI can be summarized as follows. The pressure and the convection have opposite effects during the development of KHI. However, their combined effect is positive in these vortices. This causes the kinetic energy to be transported from mean flow into the smaller scale flow structures inside the vortices. Due to the nonlinear effect of the quadratic term, the energy is further transported into higher wavenumbers, which would lead to high order instability and, finally, generate even smaller scale structures.

5.2 Vortex pairing

Another interesting phenomenon shown in Fig. 17b is that there are regions with large negative \( T_{qp,k} \) at the upstream locations of the vertices. In these regions, the energy is fed to the mean flow. This inverse cascade of the energy indicates that the flow fields in front of the vortices are basically vortex-driven. The interaction between the mean flow and these vortices will eventually cause the third stage, which is characterized by the vortex merging process.

For the vortex-driven flow, the velocity induced by a vortex affects the motion of other vortices, and as a result, some vortices begin to merge. This process has been studied by many researchers. For example in Ref. [69], Rikanatiu et al. developed a large scale statistical model to predict the merging process. In this model, the merger of vortices is achieved by the occasional “pairing” between a vortex and its neighboring vortex. The merged vortex can be further paired with one of its neighbors, triggering the formation of large mixing structures. Typical vortex trajectories can be shown in Fig. 18. This phenomenon can be also found in Fig. 11. For example, in Fig. 11a (\(t=0.500\,\hbox { ms}\)), there are more than ten vortices along the interface. In Fig. 11c (\(t=1.000\,\hbox { ms}\)), they have evolved into three large vortical regions and in Fig. 11d (\(t=1.500\,\hbox { ms}\)) there is only one turbulent region near the interface. The merging of KH vortices creates the so-called turbulent mixing zone [69]. These can be observed in Fig. 19, where the density contours of \({\mathrm {SF_6}}\) and corresponding TKE spectrum in the span-wise direction at four different instants are shown. At the early stage of KHI shown in Fig. 19a, the broadband spectrum of TKE lies only in the KH billows. However, as depicted in Fig. 19b–d, the region of the broadband spectrum of TKE is extended during the merging process of KH billows, which implies that the merging process does extend the regions with broadband scales of flow structures and enhance the overall mixing of fluids. Therefore, the merger of the KH vortices plays an important role in the formation of the turbulent mixing zone.

6 Conclusions

In the present paper, a planar shock wave-driven triangular heavy gas interface is studied in the context of the high resolution FV method with MDCD reconstruction. The vortex dynamic of RMI and the turbulent mixing during the development of KHI are discussed. An extended R model is proposed to predict the circulation for the shock triangular gas–cylinder interaction flow. Several typical processes, such as the formation of primary KH billows, the development of secondary and tertiary instabilities of KHI, as well as the merging process of KH billows, are demonstrated. The turbulent mixing transition for present case is manifested not only by the decoupled length scales but also by the extended inertial range of the TKE spectrum at its late stage. The mechanisms leading to turbulent mixing transition for the present shock triangular gas–cylinder interaction flow are further analyzed. The investigation indicates that the KH vortices absorb the energy from the mean flow and generate flow structures with higher wavenumbers, and the merging process of KH vortices extends the regions with broadband scales of flow structures and leads the flow to an interfacial turbulent mixing finally.