3.1. Statistics

Time averaging is not applicable in this flow to collect statistics for turbulence characterization. However, the central portion of the column appears to be homogeneous in the direction of the column tilt. This portion of the flow is used in the current study for collecting statistical data. Data are averaged at the vertical plane through the column axis and in the direction of the mean post-shock tilt angle of the column. For the analysis below, the flow field was cropped vertically to retain the centre third of the column. Horizontally, the domain was cropped to include the area where the SF$_6$ mass fraction first exceeds 0.001. Figure 9 shows the cropped flow field in the vertical plane for the $30^\circ$ column at $\tau =20.7$, along with an axis indicating the direction in which averages were taken.

Figure 9.

Vertical cross-section of the cropped density field for a $30^\circ$ column used for collecting statistics at $\tau =35.7$. The dashed line indicates the mean post-shock tilt angle along which averages were taken.

Favre averaging (Favre et al. Reference Favre, Kovasznay, Dumas, Gaviglio and Coantic1976) is used when extracting statistics. In this flow representation, the velocity field is decomposed into the mean and fluctuating components: $u_i = \widetilde {u_i} + u''_i$, where $\widetilde {u_i}={\left \langle \rho u_i\right \rangle }/{\left \langle \rho \right \rangle }$ is the Favre-averaged mean velocity, and $u''_{i}$ is the Favre-averaged velocity fluctuation. The spatial average of a variable $\phi$ is defined by

(3.2)\begin{equation} \left\langle\phi\right\rangle=\frac{1}{N}\sum_i \phi_i, \end{equation}

where $N$ is the number of computational cells in the direction of the column tilt. For these simulations, $N=650$ for the statistical region described above.

Below, data are presented with respect to the $x$-coordinate normalized as

(3.3)\begin{equation} x^*(y) = \frac{x-x_0(y)}{w}, \end{equation}

where $x_0(y)$ is the location of the leading edge of the column and depends on the $y$-coordinate due to the post-shock angle, and $w$ is the width of the column defined by the locations at the leading and trailing edges where the mass fraction is 0.001.

Profiles of the mean mass fraction $\left \langle Y_s\right \rangle$ and of the mean velocity components $\widetilde {u_{i}}$ were computed at different times in the streamwise direction and are shown in figure 10. In the figure, the mean velocity components are normalized by the post-shock velocity $u_{post}$.

Figure 10.

Mean mass fraction and mean velocity profiles for a column with initial tilt $\alpha _0=30^\circ$ at different times $\tau$: 5.7 (solid line), 20.7 (dashed line), and 50.7 (dotted line).

Figure 10(a) shows that at early time, the concentration of SF$_6$ is highest on the leading edge. As time progresses, the SF$_6$ concentration becomes more uniform inside the column.

Figures 10(b)–10(d) show that the flow motion in the vertical plane of the gas column occurs mainly in the $x$- and $y$-directions. In the streamwise direction, the shock accelerates the gas near the edges of the column to a higher velocity than the gas in the interior of the column. With time, the minimum value of $u_1$ moves within the column, finally localizing between the column centre and the leading edge, as shown in figure 10(b) at $\tau = 50.7$. After that time, the profile of this velocity component tends to the post-shock velocity (the value 1.0 in figure 10b) at all locations (not shown here).

In the $y$-direction, the gas column inclination causes the dense gas to accelerate upwards almost everywhere through the column width (figure 10c). The less dense gas at the column edges moves downwards. As time progresses, the dense gas motion in the area between the column centre and its trailing edge reverses its direction, with all gas moving downwards. At $\tau = 20.7$, only the gas near the leading edge is still moving upwards. By $\tau = 50.7$, there is almost no gas motion in the vertical direction.

The velocity in the $z$-direction, $u_3$, is close to zero at all locations within the column and at all times (figure 10d). This is due to the column cross-sectional symmetry being preserved with time for the chosen sample plane.

The column inclination angle was found to affect the gas $x$- and $y$-velocity components (figure 11), but not the velocity component in the $z$-direction. Specifically, in the $x$-direction, columns with initial angles up to $\alpha _0=10^\circ$ have similar profiles (figure 11a). The gas is accelerated mostly at the column leading edge and in an area behind the trailing edge. At angle $30^\circ$, the areas of high gas velocity are more narrow, with the second peak shifting closer to the column trailing edge. At any column angle, the gas at the trailing edge accelerates less than the rest of the column when experiencing a shock.

Figure 11.

Mean (a) $x$-velocity and (b) $y$-velocity at $\tau \approx 5.7$ for columns with different initial angles: $1^\circ$ (solid line), $5^\circ$ (dashed line), $10^\circ$ (dash-dotted line), and $30^\circ$ (dotted line).

In the $y$-direction, the gas was found to accelerate upwards to a higher velocity for larger initial angles $\alpha _{1}$, as shown in figure 11(b). At any angle, the gas velocity in the $y$-direction is less than in the $x$-direction.

Overall, we can infer that the gas motion in such a flow geometry is mainly 2-D in the vertical plane, and with time becomes predominantly one-dimensional (in the $x$-direction).

Normalized Reynolds stresses are presented in figure 12 as ${\sqrt {{\mathsf{R}}_{ii}}}/{u_{post}}$, where the Favre-averaged Reynolds stresses are defined as

(3.4)\begin{equation} {\mathsf{R}}_{ij} = \frac{\left\langle\rho u_i'' u_j''\right\rangle}{\left\langle\rho\right\rangle}. \end{equation}

Figure 12.

Evolution of the Reynolds stresses with time for a column with initial tilt $\alpha _0=30^\circ$ at different times $\tau$: 5.7 (solid line), 20.7 (dashed line), and 50.7 (dotted line).

These figures show that initially, the intensities of all fluctuating velocities and of the shear stresses have a maximum near the trailing edge of the column. Over time, the peak values of all Reynolds stresses move towards the leading edge of the column ($x^*=0$), with stresses generally growing in time across the entire column.

The Reynolds number describing this flow must be defined using the most physically relevant velocity and length scales. The flow features observed and characterized in this work evolve after the shock interaction with the density interface produces baroclinic vorticity. Therefore, neither the piston velocity of the shocked flow nor the shock front speed are relevant – the patterns observed are produced by the fluctuating components of the velocity field due to the deposition of baroclinic vorticity. Here, this characteristic velocity is represented by the average magnitude of fluctuating velocity, $u_{turb}=\sqrt {R_{11}+R_{22}+R_{33}}$. The appropriate length scale can be determined from the same considerations. According to Richtmyer (Reference Richtmyer1960), the initial (linearized) perturbation amplitude growth $a(t)$ is proportional to the instability wavenumber $\kappa$ and to the initial instability amplitude. For a fixed amplitude, the instability growth is faster for higher wavenumbers (and smaller wavelengths). The feature that is associated with the highest initial amplitude and the lowest wavelength in the centreline plane is the heavy-gas column diameter $\delta _0$, and in the vertical plane it is the KH wavelength $\lambda _{KH}$ (which is related to $\delta _0$ via (3.1)).

For the cases considered, the characteristic velocity is $u_{turb}=65.1\ {\rm m}\ {\rm s}^{-1}$, the mean post-shock density is $\rho _{post}=3.6\ {\rm kg}\ {\rm m}^{-3}$, and the mean mixture viscosity is $\mu _{mix}=2.65\times 10^{-5}\ {\rm kg}\ {\rm m}^{-1}\ {\rm s}^{-1}$. For the characteristic length $\delta _0$, the Reynolds number $Re_\delta = {\rho _{mix}\, u_{turb} \, \delta _0}/{\mu _{mix}}$ can be estimated as $\sim$90 000. This is perhaps the most realistic estimate, because it is associated with the scale that also produces the strongest vorticity deposition in the form of counter-rotating vortex pairs in the centreline plane. In direct experimental measurements of the flow field in a similar geometry (periodically perturbed gas curtain with wavelength $\lambda =6\ {\rm mm}$) but at an appreciably lower Mach number 1.2, the Reynolds number varied in the range $10\,000 < Re < 22\,500$ (Prestridge et al. Reference Prestridge, Rightley, Vorobieff, Benjamin and Kurnit2000).

However, the present study is concerned with the flow evolution in the vertical plane, so the scale $\lambda _{KH}$, varying in our simulations from $0.66\ {\rm mm}$ to $1.68\ {\rm mm}$, becomes relevant. An estimate for the Reynolds number based on such a length scale, $Re_\lambda = {\rho _{mix}\, u_{turb}\,\lambda _{KH}}/{\mu _{mix}}$, is in the range between 6000 and 17 000.

Information about the Reynolds number range can be used further to estimate characteristic length scales for viscous and mass diffusion processes. For viscous processes, the Taylor and Kolmogorov length scales can be approximated by $\lambda _{T} = \lambda _{KH} \, Re_{\lambda _{KH}}^{-1/2}$ and $\eta = \lambda _{KH} \, {Re}_{\lambda _{KH}}^{-3/4}$, where $\lambda _{T}$ is the Taylor microscale, and $\eta$ is the Kolmogorov microscale. For the range of Reynolds numbers ${Re}_{\lambda _{KH}}$ identified above (going with the more conservative estimate), this results in $\lambda _{T}=10\ \mathrm {\mu } {\rm m}$ to $40\ \mathrm {\mu } {\rm m}$, and $\eta =1\ \mathrm {\mu } {\rm m}$ to $10\ \mathrm {\mu } {\rm m}$. The grid spacing is ${{\rm d}x}=40\ \mathrm {\mu } {\rm m}$, which indicates that the viscous dissipation range may be partially resolved, so viscous effects must be accounted for.

The scalar dissipation length scale is relevant to the mass diffusion process and can be computed as $\eta _{D} = \eta \, {Sc}^{-3/4}$, where ${Sc}$ is the Schmidt number (Sreenivasan Reference Sreenivasan2019). The Schmidt number is defined as ${Sc}={\mu _{mix}}/({\rho _{mix} \, D})$, where $D=2.23\times 10^{-6} \ {\rm m}^2\ {\rm s}^{-1}$ is the mean diffusion coefficient between air and SF$_6$ at the post-shock temperature and pressure, resulting in $Sc=2.8$. These estimates give a range for the mass diffusion length scale $\eta _{D}$ between $0.5\ \mathrm {\mu } {\rm m}$ and $5\ \mathrm {\mu } {\rm m}$. This is smaller than the grid resolution ${{\rm d}x}=40\ \mathrm {\mu } {\rm m}$, so the length scales at which mass diffusion processes occur are not resolved. Notably, the Péclet number ${Pe}={Sc}\,{Re}_{\lambda _{KH}}$ ranges from 17 000 to 48 000, which is much larger than unity, indicating that the effects of mass diffusion occur on a significantly larger time scale than those of the advective transport. For these reasons, the effects of mass diffusion are not included in this study and can be disregarded safely on the time scales of interest.

3.2. Anisotropy

In this subsection, the flow anisotropy is studied. For this purpose, the anisotropy tensor is defined as

(3.5)\begin{equation} b_{ij}=\frac{{\mathsf{R}}_{ij}}{{\mathsf{R}}_{kk}}-\frac{1}{3}\,\delta_{ij}, \end{equation}

where $\delta _{ij}$ is the Kronecker delta, and ${\mathsf{R}}_{ij}$ is the Favre-averaged Reynolds stress tensor defined above.

The diagonal components of the anisotropy tensor give an estimate of the contribution of each fluctuating velocity component to turbulent kinetic energy. These values range from $-1/3$, corresponding to zero energy from that component, to $2/3$, corresponding to all energy coming from that component. If all three diagonal components of the anisotropy tensor $b_{ij}$ are zero, then the flow is isotropic.

Figure 13 shows diagonal components of the anisotropy tensor for different initial column angles at approximately the same time $\tau = 85.1$. Turbulent energy is distributed non-uniformly through the column at each angle. For the smaller angles (${\leq }10^\circ$), the flow is anisotropic throughout, dominated by the streamwise velocity fluctuations near the leading and trailing edges of the column, and by the spanwise fluctuations in the column interior. For the $30^\circ$ column, the flow becomes more isotropic in the interior of the column, with the transverse and vertical fluctuations having equal contributions to turbulent kinetic energy.

Figure 13.

Anisotropic tensor components for columns with initial angles (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$. The components are $b_{11}$ (solid line), $b_{22}$ (dashed line) and $b_{33}$ (dotted line).

In addition to examining the components, invariants of the anisotropy tensor can be plotted to examine the turbulence state over time. The anisotropy tensor invariants $\eta$ and $\xi$ are defined as

(3.6)$$\begin{gather} 6\eta^{2}=b_{ij}b_{ji}, \end{gather}$$

(3.7)$$\begin{gather}6\xi^{3}=b_{ij}b_{jk}b_{ki}, \end{gather}$$

where $b$ is the anisotropy tensor defined in (3.5), and repeated indices indicate summation. For realizable turbulence, $\eta$ and $\xi$ fall within the Lumley triangle, with the sides and vertices of the triangle representing special states of the Reynolds stress tensor (Pope Reference Pope2000).

Anisotropy-invariant maps are plotted for each column initial angle in figure 14. Figure 14(a) provides an overview of the Lumley triangle with an invariant map at three locations ($x^{*}=0.1,0.5,0.9$) within the $30^{\circ }$ column. For all cases, the trajectories are contained within only a small portion of the Lumley triangle, so figures 14(b–e) are restricted to the region indicated by the dotted box in figure 14(a). Arrows indicate the direction of the invariant trajectories with time.

Figure 14.

Reynolds-stress-invariant maps with (a) an overview of the Lumley triangle and (b–e) columns with different initial angles for three locations within the column: (b) $1^\circ$, (c) $5^\circ$, (d) $10^\circ$ and (e) $30^\circ$. Circles indicate $x^{*}=0.1$, triangles indicate $x^{*}=0.5$, and squares indicate $x^{*}=0.9$.

For each case, the initial values of $\eta$ and $\xi$ are near the limit of single-component turbulence as fluctuations are initially present in only the $x$-direction (figure 13). As time progresses, the flow behaviour changes in the direction of isotropy while maintaining a state close to that of axisymmetric turbulence.

3.3. Mixedness

The molecular mixedness ratio over time can also be compared for columns with different initial angles. In the current study, molecular mixedness is defined as

(3.8)\begin{equation} \theta =\frac{\left\langle\chi_s(1-\chi_s)\right\rangle_V}{\left\langle\chi_s\right\rangle_V\left\langle(1-\chi_s)\right\rangle_V}, \end{equation}

where $\left \langle {\cdot }\right \rangle _V$ is the volumetric mean defined below, and $\chi _s$ is the mole fraction of SF$_6$. The mole fraction is given by $\chi _s={n_{s}}/({n_{s}+n_{a}})$, where $n_{i}={\rho Y_{i}}/{M_{i}}$ is the molar count of species $i$ in a given grid cell with mass fraction $Y_{i}$ and molecular mass $M_{i}$. The volumetric mean is defined as

(3.9)\begin{equation} \left\langle\phi\right\rangle_V = \frac{1}{N_xN_yN_z}\sum_i\sum_j\sum_k \phi_{ijk}, \end{equation}

where $N_x$, $N_y$ and $N_z$ are the numbers of grid cells in the $x$-, $y$- and $z$-directions, respectively, and $\phi _{ijk}$ is the value at the grid cell with coordinates $(i,j,k)$.

A value $\theta =1$ means that the flow is mixed as much as possible within the volume, which means that the SF$_6$ is distributed uniformly.

Figure 15 shows the molecular mixedness ratio over time for different initial column angles. For each angle, the column has initial molecular mixedness $\theta =0.62$ due to the diffuse interface across the column defined by the initial conditions (see figure 2). Over time, the mixedness tends to value 1 for each initial angle, while the initial angle has very little effect on the mixing rate. At times beyond $\tau =60$, large-scale structures begin to impact the statistical region, and this may affect the mixedness behaviour. The apparent deviation for the $10^{\circ }$ column near $\tau \approx 0$ is a numerical artefact due to the presence of the shock wave within the statistical region under consideration that happens in this particular case. The shock is not parallel to the column tilt angle and disrupts the averaging procedure in (3.8). In other cases, solution data files do not capture the shock wave within the statistical region due to the I/O frequency of the simulation.

Figure 15.

Volume mean molecular mixedness ratio of SF$_6$ over time for different initial column angles.

3.4. Spectra

Figure 16 shows energy spectra at different times for heavy-gas columns with various initial tilt angles. Here, spectral energy is defined as $E=\left \langle \frac {1}{2}\widehat {\nu _{i}}^*\widehat {\nu _{i}}\right \rangle$, where $\nu_i = \sqrt {\rho }\,{u''_{i}}$, $\hat {({\cdot })}$ is the one-dimensional Fourier transform in the homogeneous direction, and $({\cdot })^*$ is the complex conjugate. The non-dimensional time $\tau$ depends on $t_0$, which itself depends on initial tilt angle $\alpha _0$. Simulation data files may not capture $t_0$ exactly for each angle, resulting in slightly different values of $\tau$ for each solution file set, so spectra for different initial column angles are compared at close, but non-equal times.

Figure 16.

Mean turbulent kinetic energy spectra in the vertical plane at different times for columns with initial angles (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

At every angle, an apparent power-law scaling may be observed between wavenumbers $\kappa =30$ and $\kappa =100$ (indicated by the arrows in figure 16). These wavenumbers correspond to length scales $0.4$ mm and $0.12$ mm, respectively. The values for the power-law exponent $p$ were computed by fitting a line through the apparent power-law region of the energy spectra. These values are presented in table 3. These power-law exponents are all close to $-1$ at early times, but decrease slightly in magnitude over time for all considered angles. These values are in agreement with the power-law scalings observed by Olmstead et al. (Reference Olmstead, Wayne, Simons, Monje, Yoo, Kumar, Truman and Vorobieff2017a) for scalar structure functions. In ideal Kolmogorov turbulence, exponents of the energy spectrum and the spectrum of a diffusive scalar (passive or even reactive) should be equal, provided that the same conditions are met as those necessary for the equivalence between spectral and structure function representation of the flow (Monin & Yaglom Reference Monin and Yaglom2013).

Table 3.

Energy spectra power-law exponents at different times for different initial column angles.

3.5. Structure functions

To examine further the power-law behaviour of this flow, second-order, scalar structure functions were assembled for the SF$_6$ mass fraction in a manner similar to those presented in Olmstead et al. (Reference Olmstead, Wayne, Simons, Monje, Yoo, Kumar, Truman and Vorobieff2017a):

(3.10)\begin{equation} {Y_s}_2(r)=\left\langle[Y_s(\boldsymbol{x}) - Y_s(\boldsymbol{x}+\boldsymbol{r})]^2\right\rangle, \end{equation}

where $r=|\boldsymbol {r}|$ is the magnitude of radius vector $\boldsymbol {r}$, and $\langle {\cdot } \rangle$ denotes averaging over all pairs of points in the same region as the statistics defined previously. The structure function computations have quadratic efficiency, $O(n^2)$, and can be prohibitively costly for high-resolution datasets. An algorithm was developed for the current study to produce structure function data that utilized the Kokkos framework (Edwards, Trott & Sunderland Reference Edwards, Trott and Sunderland2014) to take advantage of the performance benefits of GPU systems.

Structure functions evolving in time are shown in figure 17 for different initial gas column angles. For reference purposes, two lines are added to each plot: one corresponds to the power-law exponent $2/3$ that is expected theoretically in homogeneous isotropic turbulence (Celani et al. Reference Celani, Cencini, Vergassola, Villermaux and Vincenzi2005; Monin & Yaglom Reference Monin and Yaglom2013), and the other corresponds to the power law with the exponent close to unity observed in some experimental runs of Olmstead et al. (Reference Olmstead, Wayne, Simons, Monje, Yoo, Kumar, Truman and Vorobieff2017a). As the figure demonstrates, numerical results resemble a power law with the exponent close to unity only at the inclination angle $\alpha _0=30^\circ$ at early times, and for large scales only. There is no single power-law exponent fitting all scales at any given inclination angle at any time. Overall, the results are remarkably similar to the experiment (figure 3 of Olmstead et al. Reference Olmstead, Wayne, Simons, Monje, Yoo, Kumar, Truman and Vorobieff2017a): the plots can be interpreted as showing the transfer of energy from large scales (here, the integral scale would be related to the geometry of the initial conditions – representative gas column size after compression) to small scales, but an inertial range spanning multiple orders of magnitude does not develop.

Figure 17.

Structure functions of mass fraction at various initial inclination angles: (a) $\alpha _0=1^\circ$, (b) $\alpha _0=5^\circ$, (c) $\alpha _0=10^\circ$ and (d) $\alpha _0=30^\circ$.

This is not surprising as, strictly speaking, this flow is transitioning to turbulence (at least in the phenomenological sense) in a non-canonical case that is distinct from the homogeneous, isotropic, fully developed classical Kolmogorov turbulence.

First, the entire energy input driving the turbulent transition is provided by the shock interaction with density interfaces over a finite time. Accordingly, the spectral properties of kinetic energy fluctuations in RMI-driven mixing differ from those of Kolmogorov turbulence (Abarzhi & Sreenivasan Reference Abarzhi and Sreenivasan2022).

Second, it is important how the flow is driven on the injection scale. For the flows considered here, a reasonable assumption is that the driver is shear on several scales (dictated by the initial conditions geometry) due to RMI. A recent theory of shear-driven turbulence suggests that the second-order velocity structure function is affected by both the isotropic ($r^{2/3}$) and non-isotropic ($r^{4/3}$) contributions (Kumar, Meneveau & Eyink Reference Kumar, Meneveau and Eyink2022), whose respective roles may also be scale- and time-dependent. This could explain some of the behaviours observed in figure 17. A different way to look at decaying turbulence involves considering how the effective length scale varies with time, producing an unsteady cascade (Goto & Vassilicos Reference Goto and Vassilicos2016). A somewhat similar approach in an earlier theory (George Reference George1992) considers the possibility that a universal scaling for spectra may exist not for the wavenumber $\kappa$, but for a different object ($\kappa l$, where $l$ is the time-dependent length scale, in the George theory). The best theoretical approach to the flow considered in this work could be one taking into consideration the influence of the initial conditions (irrelevant for the fully developed turbulence case) on scalings. Whether the turbulence is affected by how it is driven in time, in space, or both, the appearance of the spectra (in terms of $\kappa$) or structure functions ($r$) may change.

Third, there is the possibility of a mismatch between numerics and experiment due to diagnostics. In experimental PLIF results, imaging shows the laser-induced fluorescence intensity from the tracer gas that usually closely follows mass concentration; however, possibilities for several diagnostic-related discrepancies remain (Melton & Lipp Reference Melton and Lipp2003). An example of such a discrepancy is discussed in § 3.6. Some subtler differences between experiment and numerics could also be attributable to differences in initial conditions: in the experiments, the diffuse gas column always has small perturbations that the numerical initial conditions do not attempt to reproduce. In a sense, this would mean that in the experiments, there is an additional initial contribution to disorder in the flow.

For possible future studies, it might be worthwhile to consider the approach described by Vorobieff et al. (Reference Vorobieff, Mohamed, Tomkins, Goodenough, Marr-Lyon and Benjamin2003), where ensembles of velocity fields from multiple experimental realizations were collected for the same experiment with nominally identical initial conditions. This made it possible to produce ensemble-averaged velocities and then extract velocity fluctuation fields for each individual experiment. Notably, while the structure functions for the full velocity fields manifested steeper slopes, the fluctuation field slopes were closer to $2/3$. In numerics, small fluctuations could be added to the initial conditions to produce similar ensembles (Palekar et al. Reference Palekar, Vorobieff and Truman2007).

3.6. Probability density functions

In this subsection, we present probability density functions (p.d.f.s) for mass fraction, vorticity components and velocity components in order to further characterize the flow development.

To construct discrete p.d.f.s, a bin size is defined to be $\Delta \phi =(\phi _{max}-\phi _{min})/N_b$ for some quantity $\phi$, where $N_b$ is the number of bins. A bin count $N_b=96$ was selected to produce smooth p.d.f. profiles with an adequate number of samples within each bin. Values of $\phi$ are then distributed into the $N_b$ bins, yielding the sample count of $N_k$ per bin. The p.d.f. is then defined by $P(\phi )={N_k}/{\Delta \phi N}$, where $N$ is the total number of samples of $\phi$. This definition results in $\sum _{k=1}^{N_k}P(\phi )\,\Delta \phi =1$.

The p.d.f.s of the ${\rm SF}_6$ mass fraction $Y_s$ (figure 18) were constructed using the mass fraction data from the same region as the statistics discussed in previous subsections. That is, the vertical cross-section is cropped vertically to retain the central one-third of the column using $Y_s=0.001$ as a criterion to determine the upstream and downstream edges of the column. Within this region, $Y_s$ falls between 0.0 and 0.27 at all times for each initial tilt angle.

Figure 18.

P.d.f.s of SF$_6$ mass fraction $Y_s$ for columns with different initial tilt angles $\alpha _0$, at different times $\tau$: 5.7 (solid line), 35.7 (dashed line), and 110.8 (dotted line). Angles are (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

At early times, immediately after passage of the shock, lower values of the mass fraction ($\approx 0.0\unicode{x2013}0.15$) are approximately equally likely at all tilt angles of the gas column, while the probability of higher mass fractions decreases rapidly (figures 18a–d). At later times, the p.d.f.s take on a bell-like shape. Tables 4(a)–4(d) present the mean, standard deviation (SD), skewness and kurtosis over time for each initial column angle. These values were computed using the mean(), std(), skewness() and kurtosis() subroutines from the pandas Python library (pandas development team 2020).

Table 4.

Mass fraction p.d.f. statistics over time for different initial column angles.

With time, the distribution becomes more narrow at all considered column angles. The mean, corresponding to the peak of the distribution, decreases with time for all initial column angles and approaches nearly the same value (${\approx }0.063$) for each column angle. That is, at later times, most gas exists in a lower concentration independent of the initial gas column angle. Similar tendencies are observed for the standard deviation.

More variation is observed with time and at different column angles in the skewness and kurtosis values (table 4). As time progresses, the skewness tends towards negative values, and more so as the column angle increases. Somewhat similar dynamics are observed for the kurtosis.

Figure 19 shows p.d.f.s of the $z$-component of vorticity that were constructed in the same way as the p.d.f.s of mass fraction. As the figure demonstrates, the column angle has little effect on the p.d.f. of this parameter and how its shape changes with time. At any time, the mean vorticity value is located around zero, with the p.d.f. shape being symmetric. With time, the distribution widens. However, at the largest column angle, $30^\circ$, the range of vorticity values does not change much between $\tau \approx 35.7$ and $\tau \approx 110.8$, which explains the larger mean value at $\tau \approx 110.8$ to compare with those at the other angles at the same time.

Figure 19.

P.d.f.s of vorticity $\omega _z$ for columns with different initial tilt angles $\alpha _0$, at different times $\tau$: 5.7 (solid line), 35.7 (dashed line), and 110.8 (dotted line). Angles are (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

Plots of p.d.f.s of the $x$-component of the gas velocity are presented in figure 20. At all considered gas column angles, a bimodal distribution emerges over time with two peaks: an upper peak corresponding to $u/u_{post}=1$, and a lower peak between $0.90$ and $0.95$. The positions of the peaks are not influenced significantly by the column angle, and have similar behaviour over time for each initial column angle. However, their amplitudes vary with time and depend on the column angle. In particular, smaller column angles have a higher initial concentration of velocity around the lower peak, but over time, the higher column angles have a larger concentration of velocity around the lower peak. The opposite is observed for the upper peak.

Figure 20.

P.d.f.s of the $x$-component of velocity normalized by the post-shock velocity $u_{post}$ at different times $\tau$: 5.7 (solid line), 35.7 (dashed line), and 110.8 (dotted line). Angles are (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

P.d.f.s of the $y$-component of the gas velocity are shown in figure 21. At the smallest column angle, there is a single peak value for $y$-velocity around $v/u_{post}=0$. With time, the distribution widens slightly, with gas starting to move in the vertical direction.

Figure 21.

P.d.f.s of the $y$-component of velocity normalized by the post shock velocity $u_{post}$ at different times $\tau$: 5.7 (solid line), 35.7 (dashed line), and 110.8 (dotted line). Angles are (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

As the column angle increases, the initial distribution widens. Over time, larger initial column angles develop a bimodal distribution of velocity in the $y$-direction, with a positive and negative mean. At $\alpha \geqslant 10^\circ$, the second peak becomes obvious with time. Both peaks have similar values at $\tau =21.4$, and their locations are shifted from $0\ {\rm m}\ {\rm s}^{-1}$ to negative and positive values. At the latest time, $\tau =110.8$, only one peak remains in the p.d.f., corresponding to $\alpha = 10^\circ$, with the peak value being positive ($v/u_{post}\approx 0.003$). This again indicates gas moving upwards. At the largest column angle, $\alpha = 30^\circ$, both peaks are seen at $\tau =110.8$ (figure 21d), but the peak with the positive value is larger ($v/u_{post}\approx 0.013$), showing that more gas is moving upwards than downwards.

The component of gas velocity in the $z$-direction (shown in figure 22) maintains a narrow distribution around $w/u_{post}=0$ for all initial column angles. Over time, the distribution widens slightly for smaller initial column angles. The zero mean is due to the symmetry of the sample plane, as discussed previously.

Figure 22.

P.d.f.s of the $z$-component of velocity normalized by the post shock velocity $u_{post}$ at different times $\tau$: 5.7 (solid line), 35.7 (dashed line), and 110.8 (dotted line). Angles are (a) $1^\circ$, (b) $5^\circ$, (c) $10^\circ$ and (d) $30^\circ$.

It is informative to compare the p.d.f.s of the SF$_6$ mass fraction (figure 18) with the experimental p.d.f.s of PLIF intensity in a very similar flow (figure 16 of Olmstead et al. Reference Olmstead, Wayne, Simons, Monje, Yoo, Kumar, Truman and Vorobieff2017a). While the relationships of the overall trends for the mixing-driven p.d.f. time evolution in the flow are clear, the PLIF intensity p.d.f.s are dominated by a peak corresponding to the tracer-free (no fluorescence) flow, which is understandably absent from figure 18.