2.1 Governing equations

The governing equations describing the mixing of two fluids with different densities can be derived from the fully compressible Navier–Stokes equations with two miscible fluids (species) with full diffusion and heat flux operators, as the infinite-speed-of-sound limit (Livescu Reference Livescu2013). In non-dimensional form, they can be written as (Cook & Dimotakis Reference Cook and Dimotakis2001; Livescu & Ristorcelli Reference Livescu and Ristorcelli2007)

(2.1)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}_{,t}^{\ast }+(\unicode[STIX]{x1D70C}^{\ast }u_{j}^{\ast })_{,j}=0, & \displaystyle\end{eqnarray}$$

(2.2)$$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D70C}^{\ast }u_{i}^{\ast })_{,t}+(\unicode[STIX]{x1D70C}^{\ast }u_{i}^{\ast }u_{j}^{\ast })_{,j}=-p_{,i}^{\ast }+\unicode[STIX]{x1D70F}_{ij,j}^{\ast }+\frac{1}{Fr^{2}}\unicode[STIX]{x1D70C}^{\ast }g_{i}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}^{\ast }$ is the density, $u_{i}^{\ast }$ is the velocity in direction $i$, $p^{\ast }$ is the pressure, $g_{i}$ is the gravity (acceleration) in direction $i$, and the stress tensor is assumed Newtonian, $\unicode[STIX]{x1D70F}_{ij}^{\ast }=(\unicode[STIX]{x1D70C}^{\ast }/Re_{0})(u_{i,j}^{\ast }+u_{j,i}^{\ast }-{\textstyle \frac{2}{3}}u_{k,k}^{\ast }\unicode[STIX]{x1D6FF}_{ij})$. In incompressible VD turbulence, the divergence of velocity is not zero due to the change in specific volume during mixing and can be written as

(2.3)$$\begin{eqnarray}u_{j,j}^{\ast }=-\frac{1}{Re_{0}Sc}\ln \unicode[STIX]{x1D70C}_{,jj}^{\ast }.\end{eqnarray}$$

This relation can also be derived either from the mass fraction transport equations or from the energy transport equation as the infinite-speed-of-sound limit (Livescu Reference Livescu2013). For the binary case, mass conservation for two fluids with constant densities, $\unicode[STIX]{x1D70C}_{1}$ and $\unicode[STIX]{x1D70C}_{2}$, requires

(2.4)$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}^{\ast }}=\frac{Y_{1}^{\ast }}{\unicode[STIX]{x1D70C}_{1}}+\frac{Y_{2}^{\ast }}{\unicode[STIX]{x1D70C}_{2}},\end{eqnarray}$$

where $Y_{1}$ and $Y_{2}$ are the mass fractions of the two fluids. This relation also represents the infinite-speed-of-sound limit of the ideal gas equation of state for the mixture (Livescu Reference Livescu2013). Since $Y_{1}^{\ast }+Y_{2}^{\ast }=1$, relation (2.4) becomes a diagnostic equation for mass fractions.

The non-dimensional parameters in (2.1)–(2.3) are the computational Reynolds number, $Re_{0}$, Schmidt number, $Sc$, and Froude number, $Fr$, defined as

(2.5)$$\begin{eqnarray}\displaystyle & \displaystyle Re_{0}=\unicode[STIX]{x1D70C}_{0}L_{0}U_{0}/\unicode[STIX]{x1D707}_{0}, & \displaystyle\end{eqnarray}$$

(2.6)$$\begin{eqnarray}\displaystyle & \displaystyle Sc=\unicode[STIX]{x1D707}_{0}/\unicode[STIX]{x1D70C}_{0}D_{0}, & \displaystyle\end{eqnarray}$$

(2.7)$$\begin{eqnarray}\displaystyle & \displaystyle Fr^{2}=U_{0}^{2}/gL_{0}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{0}=(\unicode[STIX]{x1D70C}_{1}+\unicode[STIX]{x1D70C}_{2})/2$ is the reference density, and for the initial conditions in this paper it is equal to the mean density (calculated as the volumetric average $\bar{\unicode[STIX]{x1D70C}}=(1/{\mathcal{V}})\int _{{\mathcal{V}}}\unicode[STIX]{x1D70C}^{\ast }\,\text{d}{\mathcal{V}}$ due to the homogeneity of the flow), $g$ is the magnitude of the acceleration field, $\unicode[STIX]{x1D707}_{0}$ is the reference dynamic viscosity, $D_{0}$ is the diffusion coefficient, and $L_{0}$ and $U_{0}$ are the reference length and velocity scales, respectively. For the cases investigated in this paper, $D_{0}$ is constant, while the instantaneous dynamic viscosity, $\unicode[STIX]{x1D707}^{\ast }=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}^{\ast }/\unicode[STIX]{x1D70C}_{0}=\unicode[STIX]{x1D708}_{0}\unicode[STIX]{x1D70C}^{\ast }$, where $\unicode[STIX]{x1D708}_{0}$ is the reference kinematic viscosity and is constant (see table 1). This ensures that the instantaneous Schmidt number is uniform and constant during the flow evolution. For all cases considered, $Sc=1$. The mixture rule implied by the instantaneous dynamic viscosity relation is $1/\unicode[STIX]{x1D707}^{\ast }=Y_{1}^{\ast }/\unicode[STIX]{x1D707}_{1}+Y_{2}^{\ast }/\unicode[STIX]{x1D707}_{2}$, where $\unicode[STIX]{x1D707}_{1}$ and $\unicode[STIX]{x1D707}_{2}$ are the dynamic viscosities of the two fluids. Similarly, for all cases considered in the paper, $Fr=1$.

Table 1.

Parameters for the DNS cases. *Similar results for this case, with a slightly different initialization, are available through the Johns Hopkins Turbulence Database (Livescu et al. Reference Livescu, Canada, Kanov, Burns and Pulido2014).

As shown below, it is useful to further scale the results by the velocity scale $U_{r}=\sqrt{A/Fr^{2}}$ and time scale $t_{r}=\sqrt{Fr^{2}/A}$, as some of the quantities discussed collapse with this scaling. The corresponding dimensional quantities are $U_{r}^{\dagger }=U_{r}U_{0}=\sqrt{AgL_{0}}$ and $t_{r}^{\dagger }=t_{r}L_{0}/U_{0}=\sqrt{L_{0}/Ag}$. In the triply periodic case, there is no intrinsic length scale; however, to account for the initial fluid distribution and to facilitate the comparison with other flows, such as RTI and RMI, $L_{0}$ can be taken as the initial density integral scale. Using $L_{0}$, $U_{r}^{\dagger }$ and $t_{r}^{\dagger }$ for non-dimensionalization changes the non-dimensional parameters in the governing equations to $Fr^{\dagger 2}=A$ and $Re_{0}^{\dagger }=Re_{0}\sqrt{A/Fr^{2}}=\unicode[STIX]{x1D70C}_{0}L_{0}\sqrt{AgL_{0}}/\unicode[STIX]{x1D707}_{0}$.

In HVDT, due to the periodic boundary conditions, the pressure can only be determined up to a constant gradient. Thus, the mean pressure gradient needs to be specified. Similar to Livescu & Ristorcelli (Reference Livescu and Ristorcelli2007, Reference Livescu and Ristorcelli2008), the mean pressure gradient is chosen to obtain a maximally unstable flow (understood as the time derivative of the mass flux attaining its maximum absolute value):

(2.8)$$\begin{eqnarray}P_{,i}=\frac{1}{V}\left(\frac{1}{Fr^{2}}g_{i}-\langle vp_{,i}\rangle +\langle u_{i}u_{j,j}\rangle +\langle v\unicode[STIX]{x1D70F}_{ij,j}\rangle \right),\end{eqnarray}$$

where $V$ is the mean specific volume ($v^{\ast }=1/\unicode[STIX]{x1D70C}^{\ast }=V+v$). This also leads to $U_{i}=0$; hence, in this study $u_{i}^{\ast }=u_{i}$. This choice of $P_{,i}$ is consistent with previous studies of HVDT by Sandoval et al. (Reference Sandoval, Clark, Riley, Fulachier, Lumley and Anselmet1997), where similar arguments are used but under the Boussinesq approximation. Detailed derivations to this effect can be found in § 2.2 of Livescu & Ristorcelli (Reference Livescu and Ristorcelli2007). As noted in that 2007 paper, the choice of the maximally unstable flow sets the mean velocity to the value of zero; this is similar to observations by Cabot & Cook (Reference Cabot and Cook2006) and Livescu et al. (Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009) in the core region of the RTI, where $P_{,i}$ is set by the non-periodic boundary conditions. Thus, this flow has significant similarities to the RTI, as explained in Livescu et al. (Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009). The choice $P_{,i}=0$ has been made by Gat et al. (Reference Gat, Matheou, Chung and Dimotakis2017) in their acceleration-driven mixing-layer study. In that study, which had non-zero mean velocity, the vertical growth of the mixing layers is minimized, while the horizontal growth is maximized, which has analogies with vertical convection or differential heated cavities (see also Livescu Reference Livescu2020).

2.2 Computational approach and simulation cases

Equations (2.1) and (2.2), together with the divergence condition (2.3), are solved in a triply periodic domain of size $(2\unicode[STIX]{x03C0})^{3}$, using the CFDNS code, as described in Livescu & Ristorcelli (Reference Livescu and Ristorcelli2007). The spatial derivatives are evaluated using Fourier transforms, and the time advancement is performed with the variable-time-step third-order Adams–Bashforth–Moulton scheme, coupled with a fractional-time method. To minimize the aliasing errors, the advection terms are written in the skew-symmetric form.

Table 1 lists the various cases that were chosen to investigate the influence of the Atwood and Reynolds numbers on HVDT. In the nomenclature chosen for the case names, the index following the first letter, A, varies from 1 to 4, denoting the Atwood numbers, $0.05$, $0.25$, $0.5$ and $0.75$, respectively. In addition, case names include the $Re_{0}$ index from 0 to 5 corresponding to six different values of $Re_{0}$ in increasing order, $556$, $1563$, $3125$, $4000$, $10\,000$ and $20\,000$, respectively. Also tabulated is a static buoyancy Reynolds number $Re_{b0}$ defined as (Batchelor et al. Reference Batchelor, Canuto and Chasnov1992; Livescu & Ristorcelli Reference Livescu and Ristorcelli2007) $Re_{b0}=Re_{0}\sqrt{{\mathcal{L}}_{\unicode[STIX]{x1D70C}}^{3}A/Fr^{2}}$, where ${\mathcal{L}}_{\unicode[STIX]{x1D70C}}$ is the initial density integral length scale. The Taylor Reynolds number is calculated from the turbulence Reynolds number, $Re_{t}$, using the isotropic turbulence formula $Re_{\unicode[STIX]{x1D706}}=\sqrt{(20/3)Re_{t}}$, where

(2.9)$$\begin{eqnarray}Re_{t}=Re_{0}\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime }u_{i}^{\prime \prime }\rangle ^{2}/(\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D716}),\end{eqnarray}$$

and $\unicode[STIX]{x1D716}=-\langle u_{i,j}\unicode[STIX]{x1D70F}_{ij}\rangle$ is the dissipation in the turbulent kinetic energy equation.

The density field in all simulations is initialized as a Gaussian random field with top-hat energy spectrum between wavenumbers 3 and 5. After transforming into the real space, the negative values are assigned as 1, and the positive values are assigned as $(1+A)/(1-A)$; the pure-fluid densities thus yield the desired Atwood number. To ensure that the mixing layer between the pure-fluid regions is captured on the grid, a Gaussian filter is used to smoothen the density field, thereby preserving the bounds. The width of the Gaussian filter is $1.1\unicode[STIX]{x0394}x$ and is applied once for $256^{3}$ and $512^{3}$, four times for $1024^{3}$ (resulting in a total width of ${\approx}2.2\unicode[STIX]{x0394}x$), and 16 times for $2048^{3}$ resolutions (resulting in a total width of ${\approx}4.4\unicode[STIX]{x0394}x$); the density integral length scale and mixing-state parameter (defined below) are similar for all cases.

After the initialization procedure, the non-dimensional initial density integral length scale, which is calculated from the resultant density spectra by

(2.10)$$\begin{eqnarray}{\mathcal{L}}_{\unicode[STIX]{x1D70C}}=2\unicode[STIX]{x03C0}\int _{0}^{\infty }\left.\frac{E_{\unicode[STIX]{x1D70C}^{\prime }}({\mathcal{K}})}{{\mathcal{K}}}\,\text{d}{\mathcal{K}}\right/\!\int _{0}^{\infty }E_{\unicode[STIX]{x1D70C}^{\prime }}({\mathcal{K}})\,\text{d}{\mathcal{K}},\end{eqnarray}$$

is 1.3–1.4 for all cases. The initial mixing state is represented by a commonly used mixing-state parameter, $\unicode[STIX]{x1D703}$ (Youngs Reference Youngs1991; Linden, Redondo & Youngs Reference Linden, Redondo and Youngs1994), defined as

(2.11)$$\begin{eqnarray}\unicode[STIX]{x1D703}=1-\frac{\langle \unicode[STIX]{x1D70C}^{2}\rangle }{(\bar{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C}_{1})(\unicode[STIX]{x1D70C}_{2}-\bar{\unicode[STIX]{x1D70C}})},\end{eqnarray}$$

where $\bar{\unicode[STIX]{x1D70C}}$ is the mean density. The initial value of $\unicode[STIX]{x1D703}$ is $\unicode[STIX]{x1D703}_{0}=0.068{-}0.07$ for the cases considered in table 1 except for the case A1Re1 for which $\unicode[STIX]{x1D703}_{0}=0.14$.

The non-dimensional acceleration field is $g_{i}=(-1,0,0)$ and is gradually applied to the flow between $t/t_{r}=0$ and $0.1$. A fifth-order polynomial equation is used to allow the flow to have a smooth transition from rest to the accelerated phase. This is especially important at high Atwood numbers, where exceedingly small time steps would otherwise be required at the initial times for accuracy. All simulations are well resolved, with $\unicode[STIX]{x1D702}k_{max}>2$ at all times during the flow evolution, except the A3Re2 case, for which $\unicode[STIX]{x1D702}k_{max}>1.7$. Here, $\unicode[STIX]{x1D702}=\{1/[Re_{0}^{3}(\unicode[STIX]{x1D716}/\unicode[STIX]{x1D70C}_{0})]\}^{1/4}$ is the Kolmogorov micro-scale, and $k_{max}=\unicode[STIX]{x03C0}N/{\mathcal{L}}=N/2$ is the largest resolved wavenumber. All data presented from the $256^{3}$ resolution cases represent averages over 10 realizations, while data from $512^{3}$ resolution cases represent averages over three realizations. The initial conditions for different realizations were generated using different random-number seeds. Because of the computational cost, cases with higher resolutions ($1024^{3}$ and $2048^{3}$) represent only one realization.

2.3 Energy budgets

To help describe the turbulence evolution, we define below the scalar energy (half density variance), the total and turbulent kinetic energies, as well as the potential energy, and discuss their transport equations.

2.3.1 Scalar energy ($E_{\unicode[STIX]{x1D70C}}$)

Here, the scalar energy refers to the density variance and is defined as

(2.12)$$\begin{eqnarray}E_{\unicode[STIX]{x1D70C}}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D70C}^{2}\rangle .\end{eqnarray}$$

In HVDT, during the flow evolution, $\bar{\unicode[STIX]{x1D70C}}$ remains constant due to homogeneity, so that $\unicode[STIX]{x1D70C}_{,t}^{\ast }=\unicode[STIX]{x1D70C}_{,t}$ and the rate of change of scalar energy per unit volume can be calculated by multiplying (2.1) by $\unicode[STIX]{x1D70C}$. Using homogeneity, i.e. $\langle (~~)_{,j}\rangle =0$, the transport equation turns into

(2.13)$$\begin{eqnarray}\displaystyle \frac{1}{2}\langle \unicode[STIX]{x1D70C}_{,t}^{2}\rangle & = & \displaystyle -\langle (\unicode[STIX]{x1D70C}^{\ast }-\bar{\unicode[STIX]{x1D70C}})(\unicode[STIX]{x1D70C}^{\ast }u_{j}^{\ast })_{,j}\rangle =-\langle [(\unicode[STIX]{x1D70C}^{\ast }-\bar{\unicode[STIX]{x1D70C}})\unicode[STIX]{x1D70C}^{\ast }u_{j}^{\ast }]_{,j}\rangle +\frac{1}{2}\langle \unicode[STIX]{x1D70C}_{,j}^{\ast 2}u_{j}^{\ast }\rangle \nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\langle (\unicode[STIX]{x1D70C}^{\ast 2}u_{j}^{\ast })_{,j}\rangle -\frac{1}{2}\langle \unicode[STIX]{x1D70C}^{\ast 2}u_{j,j}^{\ast }\rangle \nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2Re_{0}Sc_{0}}\langle \unicode[STIX]{x1D70C}^{\ast 2}(\ln \unicode[STIX]{x1D70C}^{\ast })_{,jj}\rangle \nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{Re_{0}Sc_{0}}\langle \unicode[STIX]{x1D70C}_{,j}\unicode[STIX]{x1D70C}_{,j}\rangle .\end{eqnarray}$$

The resultant dissipation rate, $\unicode[STIX]{x1D712}=(1/Re_{0}Sc_{0})\langle \unicode[STIX]{x1D70C}_{,j}\unicode[STIX]{x1D70C}_{,j}\rangle$, is similar to the dissipation rate for a passive scalar (Livescu, Jaberi & Madnia Reference Livescu, Jaberi and Madnia2000; Daniel, Livescu & Ryu Reference Daniel, Livescu and Ryu2018).

2.3.2 Total kinetic energy ($E_{KE}$)

The total kinetic energy is defined as

(2.14)$$\begin{eqnarray}E_{KE}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\ast }u_{i}^{\ast }\rangle .\end{eqnarray}$$

The rate of change of kinetic energy per unit volume is obtained by multiplying the momentum (2.2) by $u_{i}^{\ast }~(=u_{i})$ and averaging the resulting equation:

(2.15)$$\begin{eqnarray}E_{KE,t}=\frac{g_{i}}{Fr^{2}}\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}\rangle +\langle pu_{j,j}\rangle -\langle u_{i,j}\unicode[STIX]{x1D70F}_{ij}^{\ast }\rangle .\end{eqnarray}$$

The first term on the right-hand side is the buoyancy-production term and is proportional to the mass flux defined by

(2.16)$$\begin{eqnarray}a_{i}=\frac{\langle \unicode[STIX]{x1D70C}u_{i}\rangle }{\bar{\unicode[STIX]{x1D70C}}}=\widetilde{U}_{i}-U_{i}=-\langle u_{i}^{\prime \prime }\rangle .\end{eqnarray}$$

The second (pressure-dilatation) term is the trace of the pressure-strain tensor and is the gain or loss of kinetic energy due to the work done by the change of specific volume during mixing. For the parameters considered here, this term is negligible compared to the other terms. However, since the flow is anisotropic, the components of the pressure-strain tensor (e.g. $\langle pu_{1,1}\rangle$) are not negligible and they represent the primary mechanism of redistributing kinetic energy in different directions. The last term on the right-hand side is the total kinetic energy dissipation rate, $\unicode[STIX]{x1D716}_{tot}$.

2.3.3 Favre-averaged turbulent kinetic energy ($E_{TKE}$)

The Favre-averaged turbulent kinetic energy $(E_{TKE})$ is defined as

(2.17)$$\begin{eqnarray}E_{TKE}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime }u_{i}^{\prime \prime }\rangle ={\textstyle \frac{1}{2}}(\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\ast }u_{i}^{\ast }\rangle -\bar{\unicode[STIX]{x1D70C}}a_{i}a_{i}).\end{eqnarray}$$

The terms on the right-hand side of (2.17) are the total kinetic energy ($E_{KE}$) and mean kinetic energy ($E_{MKE}$), respectively.

The transport equation for $E_{TKE}$ can be written as (Livescu & Ristorcelli Reference Livescu and Ristorcelli2007)

(2.18)$$\begin{eqnarray}E_{TKE,t}=a_{i}P_{,i}+\langle pu_{j,j}^{\prime \prime }\rangle -\langle u_{i,j}^{\prime \prime }\unicode[STIX]{x1D70F}_{ij}^{\ast }\rangle .\end{eqnarray}$$

The first term on the right-hand side is the turbulent kinetic energy production and the last term on the right-hand side is the turbulent kinetic energy dissipation rate ($\unicode[STIX]{x1D716}$). Owing to homogeneity, $\unicode[STIX]{x1D716}_{tot}=\unicode[STIX]{x1D716}=\langle u_{i,j}^{\ast }\unicode[STIX]{x1D70F}_{ij}^{\ast }\rangle$. Similarly, $\langle pu_{j,j}^{\prime \prime }\rangle =\langle pu_{j,j}\rangle$.

2.3.4 Potential energy ($E_{PE}$)

The total available potential energy ($E_{PE}$) for the triply periodic volume ($V$) is calculated as

(2.19)$$\begin{eqnarray}E_{PE}(t)=-\frac{g_{i}}{{\mathcal{V}}Fr^{2}}\int _{{\mathcal{V}}}(\unicode[STIX]{x1D70C}^{\ast }-\bar{\unicode[STIX]{x1D70C}})x_{i}\,\text{d}{\mathcal{V}}-\int _{t=0}^{t\rightarrow \infty }{\mathcal{F}}_{E_{p}^{\ast }}\,\text{d}t,\end{eqnarray}$$

where $x_{i}$ is the relative height, which varies from $0$ to $2\unicode[STIX]{x03C0}$ in the direction opposite to the acceleration vector $\boldsymbol{g}$, and ${\mathcal{F}}_{E_{p}^{\ast }}=-(g_{i}/{\mathcal{V}}Fr^{2})\int _{S}u_{j}(\unicode[STIX]{x1D70C}^{\ast }-\bar{\unicode[STIX]{x1D70C}})x_{i}\,\text{d}S_{j}$ is the flux of the potential energy through the boundaries. The change of the available potential energy can be derived as (Livescu & Ristorcelli Reference Livescu and Ristorcelli2007)

(2.20)$$\begin{eqnarray}\displaystyle E_{PE,t} & = & \displaystyle -\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{g_{i}}{{\mathcal{V}}Fr^{2}}\int _{{\mathcal{V}}}(\unicode[STIX]{x1D70C}^{\ast }-\bar{\unicode[STIX]{x1D70C}})x_{i}\,\text{d}{\mathcal{V}}\right)-{\mathcal{F}}_{E_{p}^{\ast }}\nonumber\\ \displaystyle & = & \displaystyle -\frac{g_{i}}{Fr^{2}}\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}\rangle +\frac{g_{i}\bar{\unicode[STIX]{x1D70C}}}{{\mathcal{V}}Fr^{2}}\int _{S}u_{j}x_{i}\,\text{d}S_{j},\end{eqnarray}$$

where the first term is identical to the buoyancy-production term in (2.18), but with opposite sign, and the second term is a surface integral that goes to zero after averaging the results over different realizations.

2.4 Energy conversion rates in buoyancy-driven HVDT

In RTI under constant acceleration, the energy conversion ratio has been related to the growth rate of the RTI mixing-layer width in the alpha-group study (Dimonte et al. Reference Dimonte, Youngs, Dimits, Weber, Marinak, Wunsch, Garasi, Robinson, Andrews and Ramaprabhu2004). They reported an energy conversion rate from potential to kinetic energy ($E_{KE}/\unicode[STIX]{x1D6FF}E_{PE}$) of $0.46\pm 0.04$ when $80\,\%$ of the flow is molecularly mixed ($\unicode[STIX]{x1D703}\approx 0.8$) within the mixing layer during the self-similar regime of RTI. However, this asymptotic behaviour remains an open question, as Cabot & Cook (Reference Cabot and Cook2006) found that, after a relatively flat stage, $E_{KE}/\unicode[STIX]{x1D6FF}E_{PE}$ starts to increase again using a $3072^{3}$ DNS with $A=0.5$; this change in behaviour corresponds to the end of the alpha-group simulations where the layer reaches the domain boundaries. In HVDT, since there is no self-similar stage during the growth regimes, we report the energy conversion rate separately for the two growth regimes (explosive and saturated growths) to explore $A$ and $Re_{0}$ effects on this ratio. Similar to RTI, we define $\unicode[STIX]{x1D6FD}_{KE}$ as the ratio of the change in $E_{KE}$ to the change in available $E_{PE}$ as

(2.21)$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{KE}=\frac{\unicode[STIX]{x0394}E_{KE}}{\unicode[STIX]{x0394}E_{PE}}=(E_{KE_{t2}}-E_{KE_{t1}})\left/\!\int _{t1}^{t2}\right.E_{PE,t}(t)\,\text{d}t,\end{eqnarray}$$

where $\unicode[STIX]{x0394}E_{KE}$ is the change in $E_{KE}$ and $\unicode[STIX]{x0394}E_{PE}$ is the change in available $E_{PE}$ between any two time instants chosen for analysis. We will also discuss this ratio for the turbulent kinetic energy ($\unicode[STIX]{x1D6FD}_{TKE}$) and mean kinetic energy ($\unicode[STIX]{x1D6FD}_{MKE}$). The $\unicode[STIX]{x1D6FD}$ values are calculated by setting $t_{1}$ and $t_{2}$ in (2.21) to the normalized start and end times for the corresponding regime. For example, to calculate $\unicode[STIX]{x1D6FD}_{TKE}$ for the explosive growth regime, the start time is set as $t_{1}=t/t_{r}=0.1$; while the end time $t_{2}$ is equal to the normalized time when the condition $\text{d}E_{TKE}/\text{d}t>0$ and $\text{d}^{2}E_{TKE}/\text{d}t^{2}=0$ is reached.

2.5 Transport equation for velocity–density joint mass density function (jMDF)

Turbulent mixing as occurs in HVDT is a dynamic process where both local velocities and the fluid composition are coupled and integral to the mixing process. Here, in order to investigate the coupled effects of the composition (density) and velocity fluctuations on VD mixing, the transport equation of the density-weighted velocity–density joint p.d.f. (or velocity–density jMDF) is derived. As noted by Pope (Reference Pope1985) and Haworth (Reference Haworth2010), for VD turbulence, the mass density function (MDF) is more useful compared to the p.d.f. because the resulting equations are simpler. In addition, the velocity–density jMDF is an effective way of exploring the coupled effects of composition and velocity fluctuations, which provides comprehensive information about the buoyancy-driven VDT evolution. The jMDF (${\mathcal{F}}$) is defined as

(2.22)$$\begin{eqnarray}{\mathcal{F}}_{u_{i}\unicode[STIX]{x1D70C}^{\ast }}(V_{i},R;x_{i},t)=R\,f_{u_{i}\unicode[STIX]{x1D70C}^{\ast }}(V_{i},R;x_{i},t),\end{eqnarray}$$

where $f_{u_{i}\unicode[STIX]{x1D70C}^{\ast }}(V_{i},R;x_{i},t)$ is the joint probability density function (j.p.d.f.) of the velocity and density fields.

Following Pope (Reference Pope1985), the j.p.d.f. can be calculated using the fine-grained p.d.f., $f^{\ast }$, as $f=\langle f^{\ast }\rangle =\langle \unicode[STIX]{x1D6FF}(u_{i}^{\ast }-V_{i})\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70C}^{\ast }-R)\rangle$, where $\unicode[STIX]{x1D6FF}$ is the delta function, and $V_{i}$ and $R$ are the independent sample space variables. The details of the derivation can be found in appendix A; the final version can be written as

(2.23)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}}{\unicode[STIX]{x2202}t} & = & \displaystyle -\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}V_{i}}\left[{\mathcal{F}}\left\langle -\left.\frac{p_{,i}}{\unicode[STIX]{x1D70C}^{\ast }}\right|_{V_{i},R}-\left.\frac{P_{,i}}{\unicode[STIX]{x1D70C}^{\ast }}\right|_{V_{i},R}+\left.\frac{\unicode[STIX]{x1D70F}_{ij,j}^{\ast }}{\unicode[STIX]{x1D70C}^{\ast }}\right|_{V_{i},R}+\left.\frac{1}{Fr^{2}}g_{i}\right|_{V_{i},R}\right\rangle \right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}R}[{\mathcal{F}}\langle -\unicode[STIX]{x1D70C}^{\ast }u_{j,j}^{\ast }|_{V_{i},R}\rangle ],\end{eqnarray}$$

where $\langle Q|_{V_{i},R}\rangle$ is the conditional expectation of the function $Q(\unicode[STIX]{x1D70C}^{\ast },u_{i}^{\ast };x_{i},t)$ for specific velocity ($u_{i}^{\ast }=V_{i}$) and density ($\unicode[STIX]{x1D70C}^{\ast }=R$) values. For the homogeneous case, the explicit spatial dependence drops from the averages. Thus, in homogeneous VDT, equation (2.23) is four-dimensional. The first term on the right-hand side represents the transport of ${\mathcal{F}}$ in the velocity sample space, while the second term represents the transport of ${\mathcal{F}}$ in the density sample space (Pope Reference Pope1985). The presence of the two terms indicates that any asymmetry of the density p.d.f. is carried over to the velocity p.d.f. (or vice versa). Both $g_{i}$ and $P_{,i}$ are constant everywhere within the domain. However, since the pressure term is multiplied by the specific volume ($1/\unicode[STIX]{x1D70C}^{\ast }$), it leads to significant asymmetric behaviour at high $A$.

2.6 Conditional expectations

Owing to the large density variations among different fluid regions, differential inertial forces affect mixing (Livescu et al. Reference Livescu, Ristorcelli, Petersen and Gore2010), as well as flow dynamics. Banerjee et al. (Reference Banerjee, Kraft and Andrews2010) utilized conditional expectations by using the density field as a fluid marker to study the dynamics of bubbles and spikes for their large-$A$ number RTI experiments. They found significant differences between the statistics calculated at the bubble side (defined as $\unicode[STIX]{x1D70C}<0$) compared to the spike side (defined as $\unicode[STIX]{x1D70C}>0$).

To further explore the structure of the HVDT flow, the conditional expectations of several quantities are discussed for each of the four flow regimes. These are related to the large (i.e. turbulent kinetic energy) as well as small (i.e. dissipation and enstrophy) scales. In particular, conditional dissipation highlighted in (2.23) is an important quantity in combustion models (Klimenko & Pope Reference Klimenko and Pope2003). For binary VD flows in the incompressible limit, the mass fractions do not appear explicitly in the governing equations and can be completely determined from density. Therefore, the conditional expectations of $E_{TKE}$, dissipation of $E_{TKE}$ and enstrophy ($\unicode[STIX]{x1D714}^{2}$) are evaluated with respect to the local density (i.e. $\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime }u_{i}^{\prime \prime }|_{\unicode[STIX]{x1D70C}^{\ast }=R}\rangle$).

Figure 1.

Three-dimensional visualization of the density field for A1Re0 case: (a) initially segregated patches of heavy and light fluids at $t/t_{r}=0$; (b) at intermediate times, mixing is induced by differential buoyancy forces ($t/t_{r}\approx 2$); and (c) at late time ($t/t_{r}\approx 3$), buoyancy forces decrease and this leads to decay of $E_{TKE}$.

4.1 Explosive growth

Explosive growth is initiated when the large structures in the domain start to move fast enough to generate turbulence. As the generation of $E_{TKE}$ is accelerated in this regime ($\text{d}^{2}E_{TKE}/\text{d}t^{2}>0$), there are certain similarities with the core region of the RTI mixing layer. In RTI, when the flow becomes self-similar, $h$ (the mixing-layer width) grows quadratically, with the leading-order term of the form $Agt^{2}$. Consequently, the turbulent kinetic energy variation, which can be estimated as ${\approx}1/2{\dot{h}}^{2}$, also grows quadratically. Figure 5 shows that the growth of $E_{TKE}$ during explosive growth scales as ${\approx}t^{2}$, similar to RTI.

During this regime, the amounts of pure fluids start decreasing slowly as mixing is initiated (figure 4). This decay is slow, as molecular mixing occurs mostly at the interface of the pure fluids, where stirring first develops. No significant differences between the behaviour of the pure light and heavy fluids are observed for the $A$ numbers investigated in this paper.

Figure 5.

Growth of $E_{TKE}$ during explosive growth.

Figure 6 shows the three-dimensional (3-D) evolution of the mole fraction (figure 6a) and the velocity magnitude (figure 6b) for $A=0.05$ and $A=0.75$ (low- and high-density ratios) at different time instants. At $t/t_{r}=0$, the domain consists of pure light- and heavy-fluid patches separated from each other by thin layers; the mole fraction distribution is identical for both cases. At $t/t_{r}=0.65$, when the flow experiences a period of explosive growth, the behaviour of the density field (or mole fraction) is still similar for both low and high $A$ numbers. The large structures are observed to conserve their shapes during this regime and do not attend to molecular mixing; this is consistent with the slow decrease of the volume fractions of pure fluids in figure 4. However, at $t/t_{r}=0.65$, the velocity field is very different for low and high $A$ numbers. At first glance, the velocity field is more homogeneously distributed for the low-$A$-number case; whereas for the high-$A$-number case, the largest velocity magnitudes are more concentrated within regions that are occupied by the lighter-than-average fluid.

Figure 6.

Three-dimensional visualizations of (a) the mole fraction, where blue represents the pure light fluid ($\unicode[STIX]{x1D712}_{h}=0,\unicode[STIX]{x1D712}_{l}=1$) and red represents the pure heavy fluid ($\unicode[STIX]{x1D712}_{h}=1,\unicode[STIX]{x1D712}_{l}=0$), and (b) the velocity magnitude ($\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$), where blue represents the minimum velocity magnitude and red represents the maximum velocity magnitude, for the cases $A=0.05$ (A1Re5) and $A=0.75$ (A4Re2). The top row shows 3-D contours at $t/t_{r}=0$ while bottom row displays contours at $t/t_{r}=0.65$.

4.1.1 Energy conversion rates

Using (2.21), the energy conversion rates are calculated for different $A$ and $Re_{0}$ numbers. Figures 7 and 8 show the variations of $\unicode[STIX]{x1D6FD}_{KE}$, $\unicode[STIX]{x1D6FD}_{TKE}$ and $\unicode[STIX]{x1D6FD}_{MKE}$ with $A$ and $Re_{0}$ numbers. During explosive growth, high levels of turbulence generation occur, as most of the potential energy lost gets converted to kinetic energy ($\unicode[STIX]{x1D6FD}_{KE}>90\,\%$). For cases with the same $Re_{0}$, $\unicode[STIX]{x1D6FD}_{KE}$ is similar for all $A$ numbers; however, it increases slightly with an increase in the value of $Re_{0}$. Meanwhile, $\unicode[STIX]{x1D6FD}_{TKE}$ is found to dramatically decrease upon increase of the $A$ number. As also observed in figure 4, for moderate ($0.5$) and high ($0.75$) $A$ numbers, there is a delay in $E_{TKE}$ growth compared to $E_{KE}$. This delay might be attributed to the larger inertial differences between heavy- and light-fluid regions. Initially, heavy-fluid regions may not be stirred as efficiently as the light-fluid regions due to their larger inertia and this causes a decrease in $\unicode[STIX]{x1D6FD}_{(TKE)}$ for larger $A$ numbers. In addition, the difference between $E_{KE}$ and $E_{TKE}$ is stored within $E_{MKE}$, which acts as a reservoir for $E_{TKE}$ for the subsequent regimes in the flow evolution. Furthermore, for the low-$A$-number case, the energy conversion rates tend to asymptote to constant values, while for the high-$A$-number case, they asymptote to constant values when $Re_{0}$ is increased (see figure 8). This indicates that the flow undergoes the mixing transition in terms of energy conversion rates as the effects of the increase in the $R_{0}$ become negligible.

Figure 7.

Atwood-number effects on energy conversion rates $\unicode[STIX]{x1D6FD}_{KE}$ (black line), $\unicode[STIX]{x1D6FD}_{TKE}$ (blue line) and $\unicode[STIX]{x1D6FD}_{MKE}$ (red line) during explosive growth.

Figure 8.

Reynolds-number effects on energy conversion rates $\unicode[STIX]{x1D6FD}_{KE}$ (black line), $\unicode[STIX]{x1D6FD}_{TKE}$ (blue line) and $\unicode[STIX]{x1D6FD}_{MKE}$ (red line) during explosive growth regime for (a) $A=0.05$ (blue and black lines are identical, while the red line is close to zero) and (b) $A=0.75$.

4.1.2 Atwood-number effects on p.d.f. evolutions

Figure 9.

The p.d.f. of density field for the different (a) $A$ and (b) $Re_{0}$ values at$t/t_{r}=0.65$.

Density p.d.f.

Figure 9 presents $A$- and $Re_{0}$-number effects on the density p.d.f. during explosive growth ($t/t_{r}=0.65$). In both panels, the density field is represented by the heavy fluid’s mole fraction ($\unicode[STIX]{x1D712}_{h}$); due to incompressibility, $\unicode[STIX]{x1D712}_{h}=(\unicode[STIX]{x1D70C}^{\ast }-\unicode[STIX]{x1D70C}_{1})/(\unicode[STIX]{x1D70C}_{2}-\unicode[STIX]{x1D70C}_{1})$. The p.d.f. behaves similarly for low and high $A$ numbers. In addition, the $Re_{0}$ number does not have any significant effect on these p.d.f.s. The A1Re1 case is slightly more mixed compared to the other cases due to its higher initial mixing state as described in § 2.2. Weak $A$-number dependence on density p.d.f. is consistent with the observations reported for figure 6. The mixing rates are smaller during this regime and stirring is mostly localized; as a result $A$-number effects on density p.d.f. are not prominent.

Velocity–density jMDF

As observed in figure 6, the velocity field starts to behave differently for low and high $A$ numbers right after the acceleration ramp-up; this is noticeably earlier than the density field. The flow is highly anisotropic in the vertical (accelerated) direction (Livescu & Ristorcelli Reference Livescu and Ristorcelli2008). Figure 10 shows the normalized jMDF ($\log ({\mathcal{F}}/\bar{\unicode[STIX]{x1D70C}})$) corresponding to the vertical component of the velocity field ($u_{1}/u_{r}$) for the low- and high-$A$-number cases (at $t/t_{r}=0.65$). It is observed that, for both low and high $A$ numbers, lighter-fluid regions mostly move in the opposite direction of the acceleration field, while heavier-fluid regions mostly move in the acceleration field direction. During explosive growth, the jMDF has a double-delta shape with two peak points, as a result of initial double-delta density p.d.f.; the peak points are on the negative and positive velocity sides. These two peaks indicate that there are two different velocity subdistributions within the flow. Similar to RTI, the negative side represents the velocity distribution within pure heavy-fluid regions (which are mostly going down) and the positive side represents the velocity distribution within pure light-fluid regions (which are mostly going up). This is also consistent with high-$A$-number experiments (Banerjee et al. Reference Banerjee, Kraft and Andrews2010), where it is reported that lighter-fluid regions mostly move in the opposite direction of the acceleration field as they are associated with the rising bubbles, and heavier-fluid regions mostly move in the same direction of the acceleration field as they are associated with the dropping spikes. Moreover, for low-$A$-number cases, the velocity distribution is almost symmetric within the different regions of the flow (for the different levels of $\unicode[STIX]{x1D712}_{h}$) and density and velocity fields are moderately correlated. However, for high-$A$-number cases, the velocity distribution within the different regions of the flow becomes highly asymmetric and the pure light-fluid regions move much faster than the pure heavy-fluid regions. Those faster motions for larger $A$ numbers can be attributed to the $P_{,i}/\unicode[STIX]{x1D70C}^{\ast }$ term in (2.23), which is asymmetric due to the specific volume variation. In the light fluid, $P_{,i}/\unicode[STIX]{x1D70C}^{\ast }$ becomes large and, since turbulence is not developed yet, the nonlinear terms do not compensate for this variation.

Figure 10.

Normalized jMDFs ($\log ({\mathcal{F}}/\bar{\unicode[STIX]{x1D70C}})$) for (a) $A=0.05$ (A1Re5) and (b) $A=0.75$ (A4Re2) cases displayed at $t/t_{r}=0.65$.

4.1.3 Conditional expectations

In this subsection, we present the conditional expectations of $E_{TKE}$, $E_{TKE}$, dissipation and enstrophy to study $A$ and $Re_{0}$ effects on the asymmetry with respect to the density field.

Conditional expectation of $E_{TKE}$

The effects of variation of $A$ and $Re_{0}$ numbers on conditional expectations of $E_{TKE}$ are shown in figure 11. For higher $A$ numbers, light-fluid regions become more stirred compared to heavy-fluid regions. This is also illustrated in the asymmetric shape of the jMDF (see figure 10). Thus, $\langle \unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime }u_{i}^{\prime \prime }|_{\unicode[STIX]{x1D70C}^{\ast }=R}\rangle$ within the lightest regions (${\sim}\unicode[STIX]{x1D70C}_{1}$) is larger than $E_{TKE}$ for the high $A=0.75$ case. The $Re_{0}$ value has a relatively small effect on the distribution of the $E_{TKE}$ within the different regions of the flow during explosive growth. In addition, the $E_{TKE}$ values are larger in the lighter-fluid regions compared to the heavier-fluid regions, irrespective of the Reynolds number. This effect is the largest for the $A=0.75$ cases. The larger $E_{TKE}$ values may also lead to earlier mixing transition in those regions. This is discussed in detail below.

Figure 11.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ during explosive growth regime ($t/t_{r}=0.65$).

Conditional expectation of $E_{TKE}$ dissipation

The conditional expectation of the dissipation of $E_{TKE}$ is plotted in figure 12. For the lowest-$A$-number case, dissipation takes its highest values around the mean density. However, for the larger-$A$-number cases, the peak of dissipation moves to lighter-fluid regions. This observation is consistent with figure 10, where we observed the velocity distribution to be much wider in the lighter-fluid region, thereby causing larger velocity gradients within the region.

Increasing $A$ and/or $Re_{0}$ increases the local values more than statistics based on the whole volume. When local values become much larger than the average value, these are usually called ‘extreme events’ within the flow. For the same $Re_{0}$ values, the maximum local average (of dissipation) is larger compared to cases with lower $A$ numbers (see figure 12a). Moreover, for the A1Re5 case, $E_{TKE}$ dissipation within the fully mixed regions is more than 10 times larger than its volume average; while for the A1Re1 case, $E_{TKE}$ dissipation within the fully mixed regions is only three times larger than its volume average (see figure 12b). These larger local values indicate that both increasing $A$ and/or $Re_{0}$ numbers may increase the probability of occurrence of ‘extreme events’ within the flow.

Figure 12.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ dissipation during explosive growth regime ($t/t_{r}=0.65$).

Conditional expectation of enstrophy

The conditional expectations of the components of the enstrophy ($\unicode[STIX]{x1D714}^{2}$) in vertical and horizontal directions and the total enstrophy are plotted in figures 13(a) and 13(b), respectively. During this regime, the enstrophy behaves very similarly to dissipation, and becomes asymmetric at larger $A$ numbers. In addition, consistent with observations reported for dissipation, the local enstrophy values become larger than the volume average values upon increasing both $Re_{0}$ and $A$ numbers. Both dissipation and enstrophy are considered indicative of small-scale behaviour (Pope Reference Pope2000); thus, the asymmetric behaviour of these quantities indicates asymmetry even at small-scale motions at large $A$ numbers ($A\geqslant 0.5$). It is also observed (see figure 13a) that the vertical component of the enstrophy is much smaller than horizontal components; for example, for the A1Re1 and A4Re1 cases, $\unicode[STIX]{x1D714}_{h}^{2}$ are 10 and five times larger than $\unicode[STIX]{x1D714}_{v}^{2}$, respectively. This implies that the flow is anisotropic even at small scales. This aspect is discussed in the spectral evolution section (§ 5).

Figure 13.

Atwood- and Reynolds-numbers effects on conditional expectation of (a) vertical ($\unicode[STIX]{x1D714}_{v}^{2}=\unicode[STIX]{x1D714}_{1}^{2}$) and horizontal ($\unicode[STIX]{x1D714}_{h}^{2}=(\unicode[STIX]{x1D714}_{2}^{2}+\unicode[STIX]{x1D714}_{3}^{2})/2$) enstrophy components and (b) total enstrophy during regime (I) explosive growth ($t/t_{r}=0.65$).

4.2 Saturated growth

The time instant when $E_{TKE}$ growth starts to slow down (such that $\text{d}^{2}E_{TKE}/\text{d}t^{2}<0$) is chosen as the onset of the saturated growth regime. The 3-D evolution of the heavy-fluid mole fraction and the velocity during saturated growth are plotted in figure 14 for the low ($A=0.05$) and high ($A=0.75$) $A$ numbers. Both density and velocity fields start to behave differently for different $A$ numbers. During this period, the stirring within the flow is no longer localized and large scales start to participate in the stirring process. The turbulent intensities become large and density gradients are increased, indicating higher mixing rates within the flow. As shown in figure 4, the amounts of pure light and heavy fluids continue to decrease, but at a rate faster than during explosive growth. In addition, $A$-number effects become prominent; for larger $A$ numbers, the amounts of pure light and heavy fluids dramatically deviate from each other, indicating different mixing rates within the different flow regions, similar to results observed by Livescu & Ristorcelli (Reference Livescu and Ristorcelli2008). This breaks the symmetry of the density p.d.f., which adds more complexity to the problem. Below, it is shown that, for high $A$ numbers, the asymptotic behaviour with respect to the Reynolds number occurs earlier within pure light-fluid regions than pure heavy-fluid regions, which is taken to signify faster mixing transition within the light-fluid regions.

Figure 14.

Three-dimensional visualization of (a) the mole fraction and (b) the velocity magnitude ($\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$) for the cases $A=0.05$ (A1Re5) and $A=0.75$ (A4Re2) are displayed at $t/t_{r}=1.75$.

4.2.1 Energy conversion rates

During saturated growth, interestingly, for the same $Re_{0}$ values, the relationship between the energy conversion rates and $A$ number is not monotonic (see figure 15). Both $\unicode[STIX]{x1D6FD}_{KE}$ and $\unicode[STIX]{x1D6FD}_{TKE}$ peak at around $A$-number values of $0.25$–$0.5$. The energy conversion rates are highest at moderate $A$ numbers. In addition, $\unicode[STIX]{x1D6FD}_{TKE}$ is greater than $\unicode[STIX]{x1D6FD}_{KE}$ (see figure 4a). The stored energy in $E_{MKE}$ is released to feed into $E_{TKE}$, which leads to $\unicode[STIX]{x1D6FD}_{KE}<\unicode[STIX]{x1D6FD}_{TKE}$ and $\unicode[STIX]{x1D6FD}_{MKE}<0$. Note that $E_{MKE}$ is decreasing during saturated growth, which is also consistent with $\unicode[STIX]{x1D6FD}_{MKE}<0$ (see figure 4). It is also found that both low- and high-$A$ energy conversion rates tend to converge by increasing $Re_{0}$, indicating that the flow is beyond the mixing transition during the saturated growth regime in terms of the energy conversion rates (see figure 16).

Figure 15.

Atwood-number effects on energy conversion rates $\unicode[STIX]{x1D6FD}_{KE}$ (black line), $\unicode[STIX]{x1D6FD}_{TKE}$ (blue line) and $\unicode[STIX]{x1D6FD}_{MKE}$ (red line) during saturated growth regime.

Figure 16.

Reynolds-number effects on energy conversion rates $\unicode[STIX]{x1D6FD}_{KE}$ (black line), $\unicode[STIX]{x1D6FD}_{TKE}$ (blue line) and $\unicode[STIX]{x1D6FD}_{MKE}$ (red line) during saturated growth regime for (a) $A=0.05$ (blue and black lines overlap and red line is close to zero) and (b) $A=0.75$.

4.2.2 Atwood-number effects on p.d.f. evolutions

Density p.d.f.

Livescu & Ristorcelli (Reference Livescu and Ristorcelli2008) found that increasing the $A$ number breaks the symmetry of the mixing rates and leads to asymmetric density p.d.f.s. In their study, the reported results are for $A$ number equal to $0.5$, with maximum domain size of $512^{3}$, which was around the mixing transition threshold. The current work builds on that study to further investigate the asymmetric behaviour of the density p.d.f. evolution; density p.d.f.s for both larger $A$ number ($A=0.75$) and larger domain sizes up to $2048^{3}$ (leading to much larger $Re_{t}$ values) are presented. Figure 17 presents density p.d.f.s; the shape of the p.d.f. is distinctly different for different $A$ numbers. For the large $A$ number, the lightest-fluid regions mix faster than the heaviest-fluid regions. The density p.d.f. exhibits peak values at lighter-than-average fluid regions; this peak shifts to even lighter-fluid regions upon increasing the $A$ number.

As discussed in § 3, in order to determine whether a certain quantity is fully developed (i.e. beyond the mixing transition), we check if the time evolution of the quantity in question becomes independent of $Re_{t}$. The authors are of the opinion that for HVDT there is no single threshold that could determine whether the flow is beyond the mixing transition and that different quantities reach this transition differently. In figure 17(b), for the lowest-$A$-number case of $A=0.05$, $Re_{0}$ has minimal effect on the p.d.f. However, for large-$A$-number cases, it is observed that only the lighter-fluid regions tend to become insensitive to an increase in $Re_{0}$. The variation of the density p.d.f. in the heavier-fluid regions is still large for the high $A=0.75$ cases (the cases A075Re3 and A075Re2), even if we compare the cases with the largest two $Re_{0}$ values. This indicates that mixing transition may occur at earlier times for the lighter-fluid regions than the heavier-fluid regions when considering the density p.d.f. This is also consistent with the conditional expectation of $E_{TKE}$ (see figure 11), which was observed to be much larger in lighter-fluid regions compared to heavier-fluid regions.

Figure 17.

The p.d.f. of density field for the different (a) $A$ and (b) $Re_{0}$ values at$t/t_{r}=1.75$.

Velocity–density jMDF

The jMDF behaviour for low- and high-$A$-number cases during the saturated growth regime is shown in figure 18. For the low-$A$-number case, the jMDF is relatively symmetric with respect to the mole fraction at small velocity magnitudes. There is a slight asymmetry near the largest velocity magnitudes; the jMDF peaks at the light-fluid side at large positive velocities and on the heavy-fluid side at large negative velocities. In addition, at each heavy-fluid mole fraction level, the jMDF remains quasi-Gaussian, with the peak moving slightly from positive to negative velocities as $\unicode[STIX]{x1D712}_{h}$ varies from $0$ to $1$. The jMDF is more complex for the high-$A$-number case. Thus, the jMDF shape is wider at small values of $\unicode[STIX]{x1D712}_{h}$. At all $\unicode[STIX]{x1D712}_{h}$, the jMDF is asymmetric and skewed towards positive velocity values, though the asymmetry decreases compared to the explosive growth regime. A longer tail for positive values of the vertical velocity indicates more extreme upward motions in the flow, similar to that observed in the explosive growth.

Figure 18.

Normalized jMDFs ($\log ({\mathcal{F}}/\bar{\unicode[STIX]{x1D70C}})$) for (a) $A=0.05$ (A1Re5) and (b) $A=0.75$ (A4Re2) cases displayed at $t/t_{r}=1.75$.

4.2.3 Conditional expectations

Conditional expectation of $E_{TKE}$

The $A$- and $Re_{0}$-number effects on the conditional expectation of $E_{TKE}$ are shown in figure 19. At low $A$, $E_{TKE}$ is mostly balanced between lighter and heavier regions of the flow. As $A$ increases at fixed $Re_{0}$ value, the conditional expectation becomes asymmetric, with larger values in low- to moderate-density regions (figure 19a). An increase in $Re_{0}$ reduces the asymmetry of conditional expectation. For the higher$A$-number cases, the conditional p.d.f. converges at the lighter-fluid side. This observation is consistent with the notion of mixing transition occurring at earlier times within the lighter-fluid regions compared to the heavier-fluid regions.

Figure 19.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ during saturated growth regime (at $t/t_{r}=1.75$).

Conditional expectation of dissipation of $E_{TKE}$

The conditional expectation of the $E_{TKE}$ dissipation remains asymmetric at high $A$ during this regime, similar to the explosive growth regime (figure 20). Local dissipation expectations (computed for the specific density values) continue to take much larger values over mean dissipation values for higher $A$ and $Re_{0}$ numbers. For example, the largest conditional expectation of $E_{TKE}$ dissipation is ${\approx}3$ for the A4Re2 case and is ${\approx}3.8$ for the A1Re5 case. However, $Re_{t}$ effects decrease during the saturated growth. Furthermore, $Re_{0}$ effects on the peak of the conditional mean are smaller for the higher-$A$-number cases. This might be attributed to the location of the peak, which is within the lighter-fluid regions, where the mixing transition occurs relatively earlier than for the fully mixed and heavier-fluid regions.

Figure 20.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ dissipation during saturated growth regime (at $t/t_{r}=1.75$).

Conditional expectation of enstrophy

The conditional expectation of enstrophy is shown in figure 21(a,b); the vertical, horizontal and total components are plotted for comparison. The conditional expectation remains strongly skewed towards the light-fluid regions for the high-$A$-number cases, similar to observations reported for the explosive growth. The main difference observed in the saturated growth (compared to explosive growth) regime is that the enstrophy becomes almost isotropic at $t/t_{r}=1.75$ at the highest $A$ number. Additionally, for the high-$A$-number cases ($A\geqslant 0.5$), $Re_{0}$ effects become weaker, similar to the behaviour of the conditional expectation of the dissipation of $E_{TKE}$. Interestingly, different rates of convergence with $Re_{0}$ are observed for the different metrics discussed in this section. For example, at $t/t_{r}=1.75$, the velocity p.d.f. is substantially converged for all the cases reported (not shown here). However, the conditional expectation of $E_{TKE}$ is weakly converged for the high-$Re_{0}$ cases, while the conditional expectations of dissipation and enstrophy are further away from convergence.

Figure 21.

Atwood- and Reynolds-numbers effects on conditional expectation of (a) vertical ($\unicode[STIX]{x1D714}_{v}^{2}=\unicode[STIX]{x1D714}_{1}^{2}$) and horizontal ($\unicode[STIX]{x1D714}_{h}^{2}=(\unicode[STIX]{x1D714}_{2}^{2}+\unicode[STIX]{x1D714}_{3}^{2})/2$) enstrophy components and (b) total enstrophy during saturated growth regime (at $t/t_{r}=1.75$).

4.3 Fast decay

The fast decay regime starts when dissipation of $E_{TKE}$ begins to overcome $E_{TKE}$ generation. Since the rate of molecular mixing is high and buoyancy forces decay rapidly, the rate of $E_{TKE}$ decay increases with time such that $\text{d}^{2}E_{TKE}/\text{d}t^{2}<0$. During this regime, similarities are observed with RTI under reversed acceleration $g\leqslant 0$. In HVDT, partially mixed fluids become well mixed within the flow and turbulence starts to decay similar to observations in RTI with acceleration reversal that makes the global flow stable with the presence of local fluid patches that are unstable due to the stirring (see figure 22). These local fluid patches assist the turbulence decay due to local (but limited) turbulence generation (Ramaprabhu et al. Reference Ramaprabhu, Karkhanis and Lawrie2013; Aslangil et al. Reference Aslangil, Banerjee and Lawrie2016). Furthermore, this regime also has some similarities with shock-driven RMI. In RMI, VD turbulence generated due to the shock decays purely as there is no buoyancy assistance. However, the VD mixing physics is similar in both flows as turbulent fluid patches with different densities mix efficiently during $E_{TKE}$ decay.

Figure 22.

Three-dimensional visualization of (a) the mole fraction and (b) the velocity magnitude ($\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$) for the cases $A=0.05$ (A1Re5) and $A=0.75$ (A4Re2) are displayed at $t/t_{r}=2.95$.

Figure 23.

The p.d.f. of the density field for the different (a) $A$ and (b) $Re_{0}$ values at$t/t_{r}=2.95$.

4.3.1 Atwood-number effects on p.d.f. evolutions

Density p.d.f.

During fast decay, the density p.d.f. is highly asymmetric for large-$A$-number cases. A closer look at the tails of the distribution shows very little amounts of pure light fluid left in the flow. However, there are still substantial amounts of pure heavy fluid within the flow (see figure 23a). The normalized skewness ($S_{k}$), which identifies the asymmetry level of the p.d.f. distribution and takes positive values for the p.d.f.s with longer right-side tails and negative values for the p.d.f.s with longer left-side tails, is defined as (Ristorcelli & Clark Reference Ristorcelli and Clark2004)

(4.1)$$\begin{eqnarray}S_{k}=\frac{\langle \unicode[STIX]{x1D70C}^{3}\rangle }{\langle \unicode[STIX]{x1D70C}^{2}\rangle ^{3/2}}.\end{eqnarray}$$

Normalized skewness takes its maximum values during the fast decay regime. The $S_{k}$ value tends to slightly increase upon increasing $Re_{0}$. For all $A$-number cases (the moderate cases are not shown), the $Re_{0}$-number effect is weak during the fast decay, indicating that this quantity reaches the fully developed stage.

Velocity–density jMDF

The jMDF variations for lowest and highest $A$ numbers are shown in figure 24. As can be seen, in contrast to the density p.d.f., the asymmetry of the jMDF at each density level tends to decrease, as $P_{,i}$ and variability in $1/\unicode[STIX]{x1D70C}^{\ast }$ are lower due to mixing. However, the jMDFs are still significantly different for low- and high-$A$-number cases. Thus, for the high-$A$ case, there are still significant large positive velocity events. Moreover, due to the existence of large amounts of pure heavy fluid for high-$A$-number cases, the jMDF peaks in two regions, one close to the mean density value and near zero vertical velocity, and the other at the heaviest fluid region and slightly negative vertical velocity.

Figure 24.

Normalized jMDFs ($\log ({\mathcal{F}}/\bar{\unicode[STIX]{x1D70C}})$) for (a) $A=0.05$ (A1Re5) and (b) $A=0.75$ (A4Re2) cases displayed at $t/t_{r}=2.95$.

Figure 25.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ during fast decay regime (at $t/t_{r}=2.95$).

4.3.2 Conditional expectations

Conditional expectation of $E_{TKE}$

During fast decay, $A$ number continues to play an important role on the behaviour of the conditional expectation of $E_{TKE}$. The light- and heavy-fluid regions have larger $E_{TKE}$ values for the lower-$A$-number cases, as these regions continue to move faster. However, for high $A$ numbers, the average $E_{TKE}$ values increase slightly within the heavier-fluid regions (see figure 25).

Conditional expectation of $E_{TKE}$ dissipation

During fast decay, the conditional mean of dissipation becomes independent of $Re_{0}$. In addition, the $A$-number effect becomes weak for the range investigated here. The energy mostly dissipates within flow regions that are fully mixed and the local values are no longer much larger than the mean values (see figure 26).

Figure 26.

Atwood- and Reynolds-numbers effects on conditional expectation of $E_{TKE}$ dissipation during fast decay regime (at $t/t_{r}=2.95$).

Figure 27.

Atwood- and Reynolds-numbers effects on conditional expectation of (a) vertical ($\unicode[STIX]{x1D714}_{v}^{2}=\unicode[STIX]{x1D714}_{1}^{2}$) and horizontal ($\unicode[STIX]{x1D714}_{h}^{2}=(\unicode[STIX]{x1D714}_{2}^{2}+\unicode[STIX]{x1D714}_{3}^{2})/2$) enstrophy components and (b) total enstrophy during fast decay regime at $t/t_{r}=2.95$.

Conditional expectation of enstrophy

In contrast to the dissipation field, the conditional enstrophy is still dependent on $A$ as shown in figure 27. Similar to the dissipation field, the local values are closer to the whole-domain average for all cases and $Re_{0}$ effects are not significant. It is also noticeable that the components of the enstrophy field are similar, which indicates small-scale isotropy during this regime.

Figure 28.

Three-dimensional visualization of (a) the mole fraction and (b) the velocity magnitude ($\sqrt{u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$) for $A=0.05$ case (A1Re5) and $A=0.75$ case (A4Re2) displayed at $t/t_{r}\approx 6$.

Figure 29.

Reynolds-number evolution for pure and buoyancy-assisted decays.

4.4 Gradual decay

After the period of fast decay, the flows settles into a late-time slow decay regime, as $\text{d}^{2}E_{TKE}/\text{d}t^{2}$ again becomes larger than zero. This regime has some similarities with the core of the mixing region at late time of the RMI and RTI under reversed acceleration ($g\leqslant 0$). Buoyancy forces continue to weaken and no pure fluids exist within the flow. In addition, as most of the fluids have been molecularly mixed ($\unicode[STIX]{x1D703}>0.95$ for all cases), the VD effects are weak during gradual decay, which makes HVDT comparable to the study by Batchelor et al. (Reference Batchelor, Canuto and Chasnov1992). Figure 2 shows that HVDT attains a non-decaying Reynolds-number behaviour during the late-time stage. An interesting comparison can be made between the case of buoyancy-assisted decay and pure decay (where $g_{i}$ is set to zero at some time instant). As can be seen in figure 29, in the case of pure decay, the $Re_{t}$ number decreases continuously. However, when the decay is assisted by buoyancy forces, the Reynolds number does not decrease. This is consistent with the theoretical analysis of Batchelor et al. (Reference Batchelor, Canuto and Chasnov1992), which highlights the importance of density fluctuations in the infrared part of the energy spectrum, even as they asymptotically decrease to zero. In this study, we aim to reach as high a $Re_{t}$ value as possible during the flow evolution, so the initial peak of the density spectrum is chosen to be at low wavenumbers. Unfortunately, this choice of initial conditions leads to insufficient infrared part in the spectra to be able to draw definitive conclusions about the decay laws of $E_{\unicode[STIX]{x1D70C}}$ and $E_{TKE}$ and the behaviour of $Re_{t}$. Shifting the peak of the initial density energy spectrum to higher wavenumbers, while maintaining $Re_{t}$ number above the mixing transition, would require much larger resolutions and is outside the scope of the current work.

During the gradual decay regime, as the turbulent kinetic energy behaviour no longer changes, the flow also reaches a self-similar behaviour. Thus, density and velocity p.d.f.s tend to reach symmetric shape for all $A$ numbers investigated here. Gradual decay has strong similarities with sufficiently developed RTI under negative (reversed) acceleration where the flow is stably stratified and is not fed by pure fluids (Aslangil et al. Reference Aslangil, Banerjee and Lawrie2016). Similarly, with HVDT during gradual decay, the flow tends to become fully mixed under the presence of weak buoyancy forces primarily due to the buoyancy differences between the partially mixed regions (Livescu & Wei Reference Livescu and Wei2012; Ramaprabhu et al. Reference Ramaprabhu, Karkhanis and Lawrie2013; Aslangil et al. Reference Aslangil, Banerjee and Lawrie2016). Gradual decay also has similarities with atmospheric and oceanic flows where the flow perturbed by different sources may decay under weak buoyancy forces.

4.4.1 Atwood-number effects on p.d.f. evolutions

Density p.d.f.

During gradual decay, as most of the fluids are mixed, the density p.d.f. is accumulated at around mean density (see figure 30). Both $A$- and $Re_{0}$-number effects are limited during this regime; however, the density p.d.f. tail is slightly longer at the heavy-fluid side for larger $Re_{0}$ and $A$ numbers.

Figure 30.

The p.d.f. of the normalized density field for different (a) $A$ numbers and (b) $Re_{t}$ numbers at $t/t_{r}\approx 6$.

Velocity–density jMDF

The jMDFs of density and velocity fields (see figure 31) become closer for the low- and high-Atwood-number cases. The slightly tilted behaviour indicates that during decay, velocity and density fields remain (weakly) negatively correlated. Heavier regions moving close to the maximum downward velocities are still present in the high-$A$-number case, while slightly larger maximum vertical velocities indicate that still the light-fluid regions continue to be more stirred by turbulence.

Figure 31.

Normalized jMDFs ($\log ({\mathcal{F}}/\bar{\unicode[STIX]{x1D70C}})$) for (a) $A=0.05$ (A1Re5) and (b) $A=0.75$ (A4Re2) cases displayed at $t/t_{r}\approx 6$.

4.4.2 Conditional expectations

During gradual decay, the conditional means of $E_{TKE}$, dissipation of $E_{TKE}$ and enstrophy become less dependent on $Re_{0}$ and $A$ numbers. For all $A$ numbers studied here, $E_{TKE}$ is slightly larger within the lightest and heaviest flow regions as buoyancy forces are still present compared to the fully mixed flow (see figure 32a). Moreover, all three quantities have values close to their means in well-mixed regions, showing the importance of these regions to the global statistics. However, a lot of variability is still observed in lighter- and heavier-than-average fluid regions. In particular, while dissipation fluctuates and tends to be lower than average in light- and heavy-fluid regions, enstrophy tends to be lower than average in heavy-fluid regions, but higher than average in light-fluid regions for larger $A$ (see figure 32b,c). Thus, even at late times, lighter-fluid pockets maintain stronger vortical motions, whereas in heavier-fluid regions vorticity can become small.

Figure 32.

Atwood- and Reynolds-numbers effects on conditional expectation of normalized (a) $E_{TKE}$, (b) $E_{TKE}$ dissipation and (c) total enstrophy during gradual decay (at $t/t_{r}\approx 6$).

5.1 Spectral energy evolution

As the flow undergoes rapid changes at early times and transitions from TKE growth to decay, it is expected that the energy spectrum follows a complex evolution. In particular, the shape of the spectrum at early times should be strongly dependent on the production term, while at late times, during gradual decay, it should relax to fully developed conditions. In the previous section, a new quadratic form of the TKE was shown to have reduced viscous effects at large scales for the cases considered here, while for the constant-viscosity case those effects should vanish at large scales. This form, i.e. $\text{TKE}^{III}$, is used throughout this section.

Figure 35.

Compensated spectra of the normalized mass flux ($a_{1}(k)k^{7/3}/(A\bar{\unicode[STIX]{x1D70C}}U_{r})$) for different $A$ numbers at (a) $t/t_{r}=1.15$ and (b) $t/t_{r}=2.35$.

Figure 36.

Compensated spectra of the normalized production term for $\text{TKE}^{III}$ option ($P_{TKE^{III}}=P_{,1}u_{1}^{\prime \prime }\unicode[STIX]{x1D70C}^{\ast }\unicode[STIX]{x1D70C}/(4\bar{\unicode[STIX]{x1D70C}}A^{2})k^{7/3}$) for different $A$ numbers at (a) $t/t_{r}=1.15$ and (b) $t/t_{r}=2.35$. Dashed lines represent negative $P_{TKE^{III}}$.

For the Boussinesq limit, the buoyancy-production term, which is proportional to the mass flux ($\unicode[STIX]{x1D70C}^{\ast }u_{1}$ from (2.15)) has been reported to have $-7/3$ slope in the inertial range (Lumley Reference Lumley1967); similar observations have also been made in a stationary version of HVDT (Chung & Pullin Reference Chung and Pullin2010). Figure 35 presents the mass flux spectra (normalized by $A\bar{\unicode[STIX]{x1D70C}}U_{r}$) for different $A$ numbers at the start of the saturated growth $t/t_{r}=1.2$ and fast decay $t/t_{r}=2.4$ regimes. The data are consistent with previous studies as the mass flux has $-7/3$ slope at intermediate scales for all $A$ numbers. At early times, the $-7/3$ slope covers a relatively wide range of scales at low $A$ number. However, upon an increase in $A$ number, the mass flux spectrum has a sharper decay at small scales and the $-7/3$ slope is restricted to a shorter range of scales. As turbulence starts to decay though, the $-7/3$ slope moves to higher wavenumbers and becomes restricted to a shorter range of scales at lower $A$ number (figure 35b). This behaviour underlies the non-monotonic dependence of the flow evolution on the $A$ number. As the $A$ number is increased, the light- and heavy-fluid regions are accelerated faster, which increases the stirring, and ultimately leads to faster molecular mixing.

For the $\text{TKE}^{III}$ definition, the production term, $P_{TKE^{III}}$, is different from the mass flux and is proportional to $P_{,1}\unicode[STIX]{x1D70C}u_{1}^{\prime \prime }\unicode[STIX]{x1D70C}^{\ast }$. Figure 36 presents the spectrum of $P_{TKE^{III}}$, normalized by $4\bar{\unicode[STIX]{x1D70C}}A^{2}$. At small $A$ numbers, $P_{TKE^{III}}$ spectral shapes are close to those of the mass flux. However, at larger $A$ numbers, the production term quickly decreases at intermediate scales and even becomes negative at very small scales. This behaviour can be related to the weighting with the instantaneous density compared to the mass flux. At large scales, both the heavy- and light-fluid regions move coherently in the direction of and opposite to gravity, respectively. Thus, $u^{\prime \prime }$ and $\unicode[STIX]{x1D70C}$ have opposite signs and, since the mean pressure gradient is negative, the contributions to the production are positive in both heavy- and light-fluid regions. However, at early times, the mixing layers between pure-fluid regions contain significant inversion regions, where $u^{\prime \prime }$ and $\unicode[STIX]{x1D70C}$ have the same sign, resulting in negative production term at small scales. As the stirring encompasses the whole flow and the pure-fluid regions vanish in the later stages of the flow development, the production term becomes positive at all scales (not shown).

At all $A$ numbers, during explosive growth, the production term dominates the spectral kinetic energy balance, and the TKE develops a $-7/3$ slope (see figure 37). This slope extends over a larger range for $A=0.75$ during this regime; however, the lower-$A$-number results again underlie the non-monotonic dependence on $A$ number.

Figure 37.

Compensated energy spectra $E(k)k^{7/3}$ for different $A$ numbers (a) during explosive growth at $t/t_{r}=0.65$ and (b) at beginning of saturated growth at $t/t_{r}=1.15$.

Subsequently, during the saturated growth, due to the enhanced molecular mixing, the production term starts to weaken and the production-to-dissipation ratio decays to a value of $1$. As the equilibration between TKE production and dissipation occurs, the kinetic energy spectrum transitions from a $-7/3$ slope at intermediate scales, to a $-5/3$ slope for all $A$ numbers (see figure 38). The fast decay represents a transition period from evolving turbulence to fully developed turbulence. While at the beginning of this regime the $-5/3$ slope extends over a large range of scales, this is not the fully developed spectral shape. Indeed, the upper intermediate scales are still evolving and reach a $-1$ slope at the end of this regime. However, the Kolmogorov mechanism dominates the lower intermediate scales of the flow and they retain the $-5/3$ slope (see figure 39). Furthermore, for the higher-resolution DNS ($1024^{3}$ and $2048^{3}$), which reach higher values of $Re_{t}$ and thus a better separation between the energetic and dissipation scales, we observe a longer range of scales that have a slope of $-5/3$ at the end of the fast decay regime. During gradual decay, the spectral shape remains very similar as seen in figure 40, indicating that the flow has reached its fully developed state. Similar results are reported by Batchelor et al. (Reference Batchelor, Canuto and Chasnov1992) for their energy spectra of the Boussinesq case as well.

Figure 38.

Energy spectra for different $A$ numbers (a) during saturated growth at$t/t_{r}=1.75$ and (b) at beginning of fast decay at $t/t_{r}=2.35$ ($E(k)k^{5/3}$).

Figure 39.

Energy spectra for different $A$ numbers during fast decay at (a) $t/t_{r}=2.95$ and (b) $t/t_{r}=3.35$.

Figure 40.

Energy spectra for different $A$ numbers during gradual decay at $t/t_{r}=6$.

Several studies of buoyancy-driven turbulence have reported flatter slopes than $-5/3$ at intermediate scales. For example, Gat et al. (Reference Gat, Matheou, Chung and Dimotakis2017) reported a $-4/3$ slope in their study of a buoyancy-driven shear flow. They attributed this flatter slope to the turbulence intermittency; however, we believe that the presence of buoyancy causes the emergence of two spectral ranges at intermediate scales, with a $-5/3$ slope at smaller intermediate scales and a flatter slope at larger scales due to the assistance of buoyancy production.

5.2 Spectral anisotropy

The Reynolds stress anisotropy tensor ($\unicode[STIX]{x1D609}_{ij}$) is defined here consistent with the TKE expression as

(5.1)$$\begin{eqnarray}\unicode[STIX]{x1D609}_{ij}=\frac{\langle (\unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime })(\unicode[STIX]{x1D70C}^{\ast }u_{j}^{\prime \prime })\rangle }{\langle (\unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime })(\unicode[STIX]{x1D70C}^{\ast }u_{i}^{\prime \prime })\rangle }.\end{eqnarray}$$

For isotropic turbulence, $\unicode[STIX]{x1D609}_{11}=\unicode[STIX]{x1D609}_{22}=\unicode[STIX]{x1D609}_{33}=1/3$, while the maximum value of each component is 1. To investigate the scale dependence of the normal stress anisotropy, the definition above is extended using the kinetic energy spectrum in each direction. Owing to the presence of energy production, it is expected that the large scales remain anisotropic. This influence extends deep into the intermediate scales (Soulard & Griffond Reference Soulard and Griffond2012). However, at sufficiently large $Re_{t}$, the spectrum should become broad enough that the influence of energy production vanishes at high enough wavenumbers, so that a classical inertial range may emerge. However, under strongly non-equilibrium conditions, this picture might not hold. In particular, it is possible that the dissipation range may become anisotropic even when intermediate scales are isotropic. Thus, Livescu & Ristorcelli (Reference Livescu and Ristorcelli2008) showed a direct connection between the large and dissipation scales during the growth of HVDT at small to moderate $A$. This finding was later confirmed in a stationary but inhomogeneous version of the flow with a non-zero mean density gradient (Chung & Pullin Reference Chung and Pullin2010). Here, we explore the Atwood- and Reynolds-numbers effects on this anisotropic behaviour of intermediate and viscous ranges during the growth and decay regimes at higher Reynolds numbers and for a larger range of $A$ numbers than Livescu & Ristorcelli (Reference Livescu and Ristorcelli2008).

Figure 41 shows the vertical component of the anisotropy tensor for different $A$ and $Re_{0}$ numbers as a function of the wavenumber during explosive growth. Owing to the much larger $E_{TKE}$ production than $E_{TKE}$ dissipation, the kinetic energy spectrum is dominated by the production term and the flow is anisotropic at all scales. Up to about $k\sim 60$, the vertical component has as much energy as the horizontal components combined, similar with the overall anisotropy of the flow. While the normal stress anisotropy decreases for larger wavenumbers, it remains significant in the viscous range, especially at $A=0.75$, where it reaches higher levels than at large scales. Increasing $Re_{0}$ has a relatively minor influence on the normal stress anisotropy, and the viscous range anisotropy extends to the smallest scales of the flow.

Figure 41.

Vertical component of the anisotropy tensor as a function of the wavenumber for different (a) $A$ and (b) $Re_{0}$ numbers during explosive growth at $t/t_{r}=0.65$. In (b), longer spectrum ranges correspond to higher-resolution and higher-Reynolds-number cases.

Furthermore, compared to the cases with lower $A$ numbers, cases with larger $A$ numbers have lower anisotropy values at large scales. This observation is consistent with the $E_{TKE}$ conversion rates, which decrease upon increase in $A$ number. The $E_{TKE}$ production-to-dissipation ($P/D$) ratio is lower for the higher-$A$-number cases, which reduces the anisotropy levels at intermediate scales.

As the flow develops during the saturated growth and the $P/D$ ratio decays, the nonlinear effects slowly start to dominate at intermediate scales, which become isotropic over all $A$- and $Re_{t}$-number cases reported in this paper (figure 42). However, there remains a persistent anisotropy in the viscous range, consistent with earlier studies (Livescu & Ristorcelli Reference Livescu and Ristorcelli2008). This anisotropy is higher at small $A$ numbers, underlying the complex relation between stirring and mixing, as described throughout this paper. There seems to be a faster decrease of the anisotropy with the Reynolds number figure 42b), as the stronger stirring at early times leads to an accelerated increase in molecular mixing. The classical RTI problem or similar configuration cases where $g>0$ are the most common examples of $P\geqslant D$ flows that exhibit viscous range anisotropy. Such results are also reported for the classical RTI problem (Livescu et al. Reference Livescu, Ristorcelli, Gore, Dean, Cabot and Cook2009) from the DNS data of Cabot & Cook (Reference Cabot and Cook2006).

Figure 42.

Vertical component of the anisotropy tensor as a function of the wavenumber for different (a) $A$ and (b) $Re_{t}$ numbers during saturated growth at $t/t_{r}=1.75$.

During decay regimes, production is smaller than the total dissipation and the flow reaches its fully developed structure. Consistently, the normal stress anisotropy extends over the whole small-scale range (not shown).

5.3 Density spectra

The density energy spectra normalized by $(A\bar{\unicode[STIX]{x1D70C}})^{2}$, $E_{\unicode[STIX]{x1D70C}}(k)/(A\bar{\unicode[STIX]{x1D70C}})^{2}$, are shown in figure 43 for different $A$ numbers. In this normalization, the density spectra collapse at large scales for all $A$ numbers. During the explosive growth, the inertia associated with stirring is not sufficient to disturb the large scales of the flow as observed in figure 43(a). The initial shapes of the large structures are preserved for all $A$ numbers, as stirring and molecular mixing are mostly localized at the interface between the light- and heavy-fluid regions. This observation is also consistent with the evolution of the mole fraction shown in figure 6. As the flow transitions to saturated growth, the top-hat shape of the spectrum flattens out due to enhanced mixing (see figure 43b). The growth stages are followed by a fast decay where the top-hat shape of the density spectrum is removed and the spectra tend to become identical for all cases (see figure 43c). During late-time gradual decay, the slope of the density spectrum at large scales was observed to follow a $k^{-1}$ power law; similar values were first reported by Lesieur & Rogallo (Reference Lesieur and Rogallo1989) for passive scalar mixing and later by Batchelor et al. (Reference Batchelor, Canuto and Chasnov1992) for HVDT under Boussinesq assumption.

Even though different mixing rates are observed within different regions of the flow during both saturated growth and fast decay regimes, the density spectra are not able to capture this asymmetric behaviour; the evolution of the density spectrum is not affected by the asymmetric behaviour of the density p.d.f.

Figure 43.

Normalized density energy spectra at (a) $t/t_{r}=0$ (IC = initial condition) and $t/t_{r}=0.65$ (explosive growth), (b) $t/t_{r}=1.75$ (saturated growth), (c) $t/t_{r}=2.95$ (fast decay) and (d) $t/t_{r}=6$ (gradual decay) for different $A$ numbers.