3.1. Wide range of length scales in the turbulent mixing layer

Figures 4 and 5 display snapshots of the instantaneous mass fraction fields for all 2-D cases, and isosurfaces of the mass fraction for 3-D cases, respectively, at the end of the simulation when the rescaled time is $\tau =4.5$ . Evidently, the mixing layer for all cases exhibits fine-scale structures and large patches. Specifically, the formation of large-scale structures in the turbulent mixing layer is attributed to the merging of bubbles and spikes. The fine-scale structures are continuously generated along the evolutionary interface between heavy and light fluids (Gauthier Reference Gauthier2017). It is noteworthy that the cases with high $A_t$ and $Ro$ (e.g. case A09R20) exhibit more fine-scale structures compared to the cases with low $A_t$ and $Ro$ (e.g. case A01R025). This is due to the fact that a higher ${\textit{Re}}_H$ is achieved for the case with higher $A_t$ and $Ro$ at the same rescaled time. The visualisations clearly demonstrate the wide range of length scales present in rotation-driven RT turbulence.

Figure 4.

Snapshots of instantaneous fields for the mass fraction of heavy fluid for 2-D cases at the rescaled time $\tau =4.5$ . Here, the radii of all panels are $2.5r_0$ .

Figure 5.

Isosurfaces of the mass fraction of heavy fluid (orange for $Y_h = 0.01$ , grey for $Y_h = 0.5$ , and purple for $Y_h = 0.99$ ) for 3-D cases at the rescaled time $\tau =4.5$ .

To quantitatively describe the wide range of length scales in rotation-driven RT turbulence, the density spectra defined by (Tritschler et al. Reference Tritschler, Zubel, Hickel and Adams2014; Ge et al. Reference Ge, Zhang, Li and Tian2020)

(3.1) \begin{equation} {E_\rho (r,k_\theta ,t)}=\frac {\widehat {\rho '}^* \widehat {\rho '}(r,k_\theta ,t)}{\Delta k_\theta }. \end{equation}

are examined. Here, $k_\theta$ is the wavenumber in the circumferential direction, and the spectra bin size is given by $\Delta k_\theta =\Delta n\,2\unicode{x03C0} /(2\unicode{x03C0} r_0)=1$ , with $\Delta n=1$ at the location of the initial interface ( $r_0=1$ ). In addition, $\widehat {\rho '}$ is the Fourier transformation of the density fluctuation, and $\widehat {\rho '} ^*$ is its complex conjugate. All the density spectra shown in figure 6 are calculated at the interface $r_0=1$ and normalised by total spectral density $E_{\textit{tot}}=\int E_\rho\, {\textrm{d}} k_\theta$ . For 2-D cases, $k_{\theta ,max} = 800$ with $N=1600$ circumferential grid points; here, $k_{\theta ,max}$ is maximum wavenumber. In contrast, for 3-D cases, $k_{\theta ,max} = 400$ with $N=800$ circumferential grid points. Moreover, the density spectra for 3-D cases are averaged in the axial direction. To reveal the influence of rotational speed on the flow structures in rotation-driven RT turbulence, we compare the spectra with the same $A_t$ , and meanwhile those spectra are evaluated at the moment when ${\textit{Re}}_H$ reaches a same value. Specifically, the ${\textit{Re}}_H$ values in figure 6(a,d) ( $A_t=0.1$ ), figure 6(b,e) ( $A_t=0.5$ ) and figure 6(c,f) ( $A_t=0.9$ ) are $11\,000$ , $14\,000$ and $21\,000$ , respectively. Notably, the widths of mixing layers $h$ for all cases at the moment shown in figure 6 are approximately of the order of $ r_0$ . Evidently, the rotation-driven RT flows cover a wide range of length scales, including the scales for initial perturbations shaded in orange, as well as other scales excited by the nonlinear interaction of bubbles and spikes. This demonstrates that a turbulent state is achieved when ${\textit{Re}}_H$ reaches the order of $10^4$ (Cook & Zhou Reference Cook and Zhou2002). Interestingly, at the same $A_t$ and ${\textit{Re}}_H$ , the energy spectra for $Ro=0.25$ are smaller than those for $Ro=1.0$ and $Ro=2.0$ in the 2-D cases within the high wavenumber range, as indicated by the blue shading. Similarly, the spectra for $Ro=0.25$ are also smaller than those for $Ro=2.0$ in the 3-D cases with the high wavenumber range at the same $A_t$ and ${\textit{Re}}_H$ . This finding suggests that strong rotation at the same $A_t$ promotes the generation of small-scale structures in rotation-driven RT turbulence. Moreover, for 2-D cases with the same $A_t$ , the spectra for $Ro = 1.0$ and $2.0$ are nearly identical. This observation is related to the dynamics of the flow. Specifically, as shown in table 3, at the same $A_t$ , the ratios of average magnitude of Coriolis force to that of centrifugal force at $r=1$ are nearly similar for the 2-D cases $Ro=1.0$ and $2.0$ . The ratios for $Ro=1.0$ and $2.0$ are significantly smaller than the ratio for $Ro=0.25$ . The above phenomena indicate that at the same $A_t$ and ${\textit{Re}}_H$ , the nearly identical spectra for $Ro=1.0$ and $2.0$ correspond to similar ratios, while a larger ratio in the case $Ro=0.25$ corresponds to a reduction in the spectrum. It is important to note that the correlation between the spectral features and the ratios of forces is merely an interesting phenomenon. The physical mechanisms underlying these spectral features require further in-depth investigation.

To further confirm the beneficial effect of rotation on the generation of small-scale structures, the ratio of total turbulent kinetic energy ( $E_t$ ) of mixing layer, representing the small-scale intensity in mixing layer, to total kinetic energy ( $E_k$ ) is examined. Here, $E_t= ({1}/{2}) \int _V \rho u_i'' u_i''\,{\textrm{d}}V$ with velocity fluctuation $u_i''$ , and $E_k = ({1}/{2}) \int _V \rho u_i u_i\, {\textrm{d}}V$ , where the integration domain $V$ corresponds to the mixing layer. Figure 7 presents the temporal evolution of the ratio of $E_t$ to $E_k$ for 2-D and 3-D cases. Clearly, $E_t/E_k$ increases as $Ro$ increases under the same values of $A_t$ , providing further evidence that strong rotation promotes the generation of small-scale structures in rotation-driven RT turbulence.

3.2. Self-similarity feature of the turbulent mixing layer

Another important flow feature of RT turbulence is self-similarity, which can be observed from the distribution of mean mass fraction across the mixing layer (Livescu et al. Reference Livescu, Wei and Brady2021; Abarzhi & Sreenivasan Reference Abarzhi and Sreenivasan2022). To confirm whether the rotation-driven RT turbulence exhibits the self-similarity, four typical cases, namely A01R025, A09R20, 3DA01R025 and 3DA09R20, are examined. These four cases are selected as they achieve the lowest or highest values of ${\textit{Re}}_H$ at the end of the 2-D and 3-D simulations, respectively. Figure 8 presents the radial distributions of the mean mass fraction of the heavy fluid at four different moments for these cases. As time progresses, the decrease of mass fraction along the radial direction becomes more gradual, indicating an increasing mixing width. Nevertheless, the distribution of mass fraction of heavy fluid exhibits a similar radial decreasing trend for each case, suggesting the self-similarity in rotation-driven RT turbulence. Therefore, the distribution of mean mass friction of heavy fluid can be represented as (Ristorcelli & Clark Reference Ristorcelli and Clark2004)

Table 3.

The ratio of the average magnitude of the Coriolis force $\langle \vert F_{\textit{cor}} \vert \rangle$ to that of centrifugal force $\langle \vert F_{\textit{cen}}\vert \rangle$ on the interface $r_0 = 1$ at the time corresponding to figure $6$ , for the 2-D cases.

Figure 6.

The density energy spectra at $r=1$ in the circumferential direction, with (a,d) $A_t=0.1$ , ${\textit{Re}}_H=11\,000$ , (b,e) $A_t=0.5$ , ${\textit{Re}}_H=14\,000$ , and (c,f) $A_t=0.9$ , ${\textit{Re}}_H=21\,000$ . The wavenumbers of initial perturbations are shaded in orange, and the wavenumbers corresponding to the small scales affected by the rotation are shaded in blue. The black arrows in each panel indicate an increase in $Ro$ , corresponding to an increase in rotation.

Figure 7.

Temporal evolution of ratios of the turbulent kinetic energy to the total kinetic energy for 2-D cases with (a) $A_t=0.1$ , (b) $A_t=0.5$ , (c) $A_t=0.9$ , and 3-D cases with (d) $A_t=0.1$ , (e) $A_t=0.5$ , (f) $A_t=0.9$ .

Figure 8.

The radial distributions of mean mass fraction of heavy fluid at four different moments for cases (a) A01R025, (b) A09R20, (c) 3DA01R025 and (d) 3DA01R20. For clarity in the presentation, the solid line $\tau =4.0$ is plotted.

Figure 9.

The distributions of mean mass fraction of heavy fluid at four different moments versus similarity variable $\xi$ for cases (a) A01R025, (b) A09R20, (c) 3DA01R025 and (d) 3DA01R20.

(3.2) \begin{equation} {\langle Y_h \rangle (r,t) }=Y_0\, F_Y(\xi ), \quad \xi =\frac {r-r_0}{h(t)}, \end{equation}

where $Y_0$ is the scaling coefficient, $F_Y(\xi )$ is the spatial similarity function, and the mixing width $h$ is defined as the sum of widths of the inward and outward mixing layers, i.e. $h=h_{\textit{in}} + h_{\textit{out}}$ .

As shown in figure 9, the distributions of mean mass fraction at different moment instances collapse nearly onto a single curve for these representative cases. This suggests that the rotation-driven RT flow achieves a turbulent state with self-similarity (Youngs Reference Youngs2013; Hillier Reference Hillier2020). Additionally, it is worth noting that the growth law $h (t )$ of the mixing width, as shown in (3.2), is of critical importance in the self-similarity of RT turbulence.

3.3. Width and mixed mass of the turbulent mixing layer

The width of the mixing layer $h$ is a critical measurement in RT turbulence, thus the temporal evolution of the mixing width in rotation-driven RT turbulence is examined and shown in figure 10. Evidently, the mixing layer growth in rotation-driven RT turbulence does not scale as $t^2$ at late time, which is dramatically different from the planar RT turbulence induced by constant external acceleration $g$ (Cook et al. Reference Cook, Cabot and Miller2004; Ristorcelli & Clark Reference Ristorcelli and Clark2004; Zhao et al. Reference Zhao, Liu and Lu2020a ). This indicates that the model of planar RT turbulence, which includes only buoyancy effects, is insufficient for predicting the growth of the mixing width in the present simulations where the Coriolis force is significant. Thus it is essential to develop a new theoretical model to predict the growth of the mixing width in rotation-driven RT turbulence, taking into account the unique mechanisms involved, such as the contribution of the Coriolis force.

Figure 10.

Temporal evolutions of the width of the mixing layer $h$ .

Figure 11.

Temporal evolutions of the normalised mixed mass $\varPsi$ .

Figure 12.

The radial distribution of $e_t$ , defined as the turbulent kinetic energy $\langle \rho u_i'' u_i''/2\rangle$ normalised by $[\rho u_i u_i/2]$ , along the similarity variable $\xi$ at $\tau =4.5$ .

Figure 13.

The radial distribution of $\psi$ , defined as the mixed mass $\langle \rho Y_h Y_l\rangle$ normalised by $[ \langle \rho \rangle \langle Y_h \rangle \langle Y_l \rangle ]$ , along the similarity variable $ \xi$ at $\tau =4.5$ .

The normalised mixed mass, which characterises the degree of fluid mixing, is another fundamental measurement of the evolution of the mixing layer, and is defined as (Zhou et al. Reference Zhou, Cabot and Thornber2016)

(3.3) \begin{equation} {\varPsi }=\frac {\int \rho Y_h Y_l\, {\textrm{d}}V} {\int \langle \rho \rangle \langle Y_h \rangle \langle Y_l \rangle\, {\textrm{d}}V}. \end{equation}

Obviously, $\varPsi$ ranges from $0$ to $1$ , corresponding to the completely segregated fluids to the completely mixed fluids. It is shown in figure 11 that the normalised mixed mass curves have valley values at early times, which is consistent with the observation in planar RT turbulence induced by a constant external acceleration $g$ . Figure 11 shows that the value of $\varPsi$ at the same $A_t$ decreases with increasing $Ro$ , indicating that strong rotation is not conducive to overall fluid mixing within the mixing layer. Furthermore, for the same $A_t$ and $Ro$ , the normalised mixed mass of 3-D cases is greater than that of 2-D cases. This observation may be due to the stretching and tilting of the vortex structure in 3-D cases, which promotes fluid mixing. It should be noted that $\varPsi =1$ at the beginning is a result of initial mixing layer settings. This phenomenon, where $\varPsi =1$ at the beginning, is also observed in the study of RT flow (Zhou et al. Reference Zhou, Cabot and Thornber2016).

The above findings indicate that at the same $A_t$ , rotation promotes the generation of small-scale structures but is detrimental to the overall mixing of fluids within the mixing layer. To provide an in-depth explanation of this phenomenon, the radial distributions of $e_t$ and $\psi$ are presented in figures 12 and 13, respectively. Here, $e_t$ and $\psi$ are defined as

(3.4) \begin{equation} e_t = \frac {\big\langle \rho u_i'' u_i''/2 \big\rangle } {[ \rho u_i u_i/2 ] }, \quad \psi = \frac {\langle \rho Y_h Y_l \rangle } {[\langle \rho \rangle \langle Y_h \rangle \langle Y_l \rangle ] }, \end{equation}

respectively. Indeed, $e_t$ represents the turbulent kinetic energy $\langle \rho u_i'' u_i''/2\rangle$ normalised by $[\rho u_i u_i/2]$ , reflecting the generation of small-scale structures at the radial position $r$ , and $\psi$ is the mixed mass $\langle \rho Y_h Y_l\rangle$ normalised by $[ \langle \rho \rangle \langle Y_h \rangle \langle Y_l \rangle ]$ , indicating the degree of mixing at the radial position $r$ . It is important to clarify that we apply a Savitzky–Golay filter function with polynomial order 1 and window size $20$ to the distributions of $e_t$ and $\psi$ in 2-D cases, as specified in Lavacot et al. (Reference Lavacot, Liu, Williams, Morgan and Mani2024). This approach helps to mitigate the influence of limited statistical samples available in 2-D RT turbulence on the characteristics of physical quantity distributions (Lavacot et al. Reference Lavacot, Liu, Williams, Morgan and Mani2024).

As shown in figures 12 and 13, higher $e_t$ correlates with greater $\psi$ , indicating that fine-scale structures promote small-scale mixing. However, as $Ro$ increases, $e_t$ becomes more concentrated in the central region of the mixing layer, resulting in a sharper distribution curve. Consequently, as the rotation increases, $\psi$ in the regions near the frontiers of the mixing layer is significantly reduced, resulting in an overall decrease in fluid mixing within the layer. Additionally, it is noteworthy that the distributions of $e_t$ and $\psi$ exhibit asymmetry relative to the initial interface in figures 12 and 13, which stems from the asymmetries in flow evolution and cylindrical geometry. On the one hand, it is widely recognised that the evolution of bubble and spike mixing regions in RT turbulence is asymmetric, resulting in an asymmetric distribution of physical quantities relative to the initial interface (Zhou et al. Reference Zhou, Cabot and Thornber2016; Zhou Reference Zhou2017b ). On the other hand, the cylindrical geometry and centrifugal force varying in the radial direction are inherently asymmetric with respect to the initial interface, leading to an asymmetric distribution of physical quantities. Thus the distributions of $e_t$ and $\psi$ are reasonably asymmetric due to the combined effects of asymmetries in flow evolution and cylindrical geometry.

3.4. Theoretical model of the mixing layer growth

Building on the approach employed by Zhao et al. (Reference Zhao, Wang, Liu and Lu2021) in cylindrical RT turbulence without rotation, a theoretical model is proposed in this subsection to predict the growth of mixing width in rotation-driven RT turbulence with the initial perturbed interface $r_0$ and angular speed of rotating system $\varOmega$ . Here, choosing the angular speed $\varOmega$ to establish the model instead of $Ro$ is for the convenience of model application. In this manner, other studies of rotation-driven RT turbulence can directly utilise the model without employing the present non-dimensionalisation approach. Since the mixing regions on both sides of the initial perturbed interface $r_0$ are asymmetrical, the inward and outward mixing layer widths, $h_{\textit{in}}$ and $h_{\textit{out}}$ , need to be modelled separately. Before establishing the model, it is essential to understand the physical mechanisms underlying the development of the mixing layer in RT turbulence. The basic mechanism of RT turbulence is the buoyancy-induced mixing of fluids (Boffetta & Mazzino Reference Boffetta and Mazzino2017). For planar RT turbulence with a constant external acceleration $g$ , the growth of the mixing width can be modelled by (Cook et al. Reference Cook, Cabot and Miller2004; Zhao et al. Reference Zhao, Wang, Liu and Lu2021)

(3.5) \begin{equation} \frac {{\textrm{d}}\dot {h}_i}{{\textrm{d}}t} = 2\alpha _i A_t g, \end{equation}

where $i=b,s$ , $h_b$ denotes the bubble mixing width, $h_s$ represents the spike mixing width, and $\alpha _i$ denotes the dimensionless coefficient. In rotation-driven RT turbulence, the inward and outward mixing layers correspond to the bubble and spike regions, respectively. The flow mechanism described by (3.5) can be outlined as follows. The growth of the bubble and spike regions can be attributed to the buoyancy force expressed by $A_t$ (denoting the density difference) and the external acceleration $g$ . Specifically, the right-hand-side term denotes the average buoyancy force acting on the fluid parcels at the frontier of the mixing layer, and the left-hand-side term represents the acceleration of the mixing region, which can also be referred to as the acceleration of fluid parcels at the frontier of the mixing layer.

Figure 14.

Ratios of the mean radial velocity at the positions of the (a) outer and (b) inner frontiers of the mixing layer to the growth velocity of the mixing layer versus the rescaled time $\tau$ .

As for rotation-driven RT turbulence, the growth of bubbles and spikes in the corresponding pure fluids can also be viewed as result of the forces acting on the fluid parcels at the frontier of the mixing layer. Inspired by previous work (Zhao et al. Reference Zhao, Wang, Liu and Lu2021), the model for rotation-driven RT turbulence can be established by analysing the acceleration of the fluid parcels at the frontier of the mixing layer, and the forces acting on them. We first need to clarify the relationship between the growth speed of the mixing layer with the radial velocity of fluid parcels at the mixing frontiers. The ratios $\left \langle u_r \right \rangle _{\textit{in}}/\dot {h}_{\textit{in}}$ and $\left \langle u_r \right \rangle _{\textit{out}}/\dot {h}_{\textit{out}}$ at the inner and outer frontiers of the mixing layer, respectively, are presented in figure 14. Evidently, the absolute values of ratios remain within a narrow range $0.9$ – $1.1$ at $\tau \gtrsim 1.5$ when the flows reach turbulent states, indicating that the interface speed can be approximated by the mean radial velocity of parcels at mixing frontiers. Therefore, based on this approximation, the interface speed can be modelled by establishing the evolution equation for the radial velocity of fluid parcels at mixing frontiers using Newton’s second law. The buoyancy force driving the parcels at the mixing frontiers in rotation-driven RT turbulence is caused by the centrifugal force and can be modelled in a form similar to (3.5), i.e. $2\alpha _iA_t\varOmega ^2 r_i$ , where $i=\textit{out},\textit{in}$ . It is worth noting that the centrifugal force varies linearly along $r$ , thus the modelled buoyancy force is related to the position of the frontier of the mixing layer $r_i$ . However, in addition to the buoyancy force, the contribution of the Coriolis force in rotation-driven RT turbulence to the growth of the mixing layer must also be taken into account. The radial component of Coriolis force acting on fluid parcels can be formulated as the term $2{\varOmega } {u}_\theta$ . In order to quantitatively assess the contribution of the Coriolis force to the growth of the mixing layer, the ratios $\left \langle u_\theta \right \rangle _{\textit{in}}/\dot {h}_{\textit{in}}$ and $\left \langle u_\theta \right \rangle _{\textit{out}}/\dot {h}_{\textit{out}}$ at the inner and outer frontiers of the mixing layer, respectively, are plotted in figure 15. Evidently, the absolute values of ratios $\left \langle u_\theta \right \rangle _{\textit{out}}/\dot {h}_{\textit{out}}$ and $\left \langle u_\theta \right \rangle _{\textit{in}}/\dot {h}_{\textit{in}}$ remain within a narrow range $0.85$ – $1.1$ at $\tau \gtrsim 1.5$ when the flows reach turbulent states. Therefore, it is reasonable to assume that the magnitudes of $\left \langle u_\theta \right \rangle$ at the inner and outer frontiers of the mixing layer are approximately equal to $\dot {h}_{\textit{in}}$ and $\dot {h}_{\textit{out}}$ during the turbulent mixing phase. Moreover, the ratios exhibit positive values at the inner mixing frontier, and negative values at the outer mixing frontier, indicating that $\left \langle u_\theta \right \rangle$ has different directions along the radial direction. To this end, the direction of the Coriolis force needs further determination, and the distributions of $\left \langle u_\theta \right \rangle$ at four different moments for four representative cases have been incorporated here. As shown in figure 16, the distributions of $\left \langle u_\theta \right \rangle$ show positive values in the inward mixing region, and negative values in the outward mixing region, indicating that the Coriolis force suppresses both the inward growth of the bubble region and the outward growth of the spike region. Hence the contribution of Coriolis force to the growth of the mixing layer in rotation-driven RT turbulence can be modelled as the term $-2\varOmega \dot {h}_i$ . It is clear that the forces in rotation-driven RT turbulence are more complex than those in cylindrical RT turbulence without rotation (Zhao et al. Reference Zhao, Wang, Liu and Lu2021). This is due to rotation, which causes buoyancy to be inhomogeneous along the radial direction, and introduces the Coriolis force.

Figure 15.

Ratios of the mean circumferential velocity at the positions of the (a) outer and (b) inner frontiers of the mixing layer to the growth speed of the mixing layer versus the rescaled time $\tau$ .

Figure 16.

The distribution of the mean circumferential component of velocity $u_\theta$ at four different moments versus the similarity variable $\xi$ for cases (a) A01R025, (b) A09R20, (c) 3DA01R025 and (d) 3DA01R20.

In addition to the dynamic factors (i.e. the buoyancy force and Coriolis force), the average acceleration of fluid parcels at the mixing frontiers also needs to be determined. The radial acceleration $a_r$ of each fluid parcel at the frontiers of the mixing layer in the cylindrical coordinate system is defined as

(3.6) \begin{equation} a_r=\frac {{\textrm{d}}u_r}{{\textrm{d}}t} - \frac {u_\theta ^2}{r}. \end{equation}

Note that the average radial velocity $\langle u_r\rangle$ at the outer and inner frontiers of the mixing layer can be approximated as $\dot {h}_{\textit{out}}$ and $-\dot {h}_{\textit{in}}$ , respectively, based on the results shown in figure 14. Moreover, the term $u_\theta ^2/r$ describes the contribution of circumferential motion to the radial acceleration of the fluid parcel. As shown in figure 15, $\left \langle u_\theta \right \rangle$ values at the positions of the inner and outer frontiers of the mixing layer have the same order of magnitude as $\dot {h}_{\textit{in}}$ and $\dot {h}_{\textit{out}}$ when the flows reach the turbulent state. Therefore, the average term $u_\theta ^2/r$ can be estimated as $\dot {h}^2_{\textit{out}}/r_{\textit{out}}$ and $\dot {h}^2_{\textit{in}}/r_{\textit{in}}$ for outward and inward mixing layers, respectively. Evidently, the acceleration in rotation-driven RT turbulence has the same form as that in cylindrical RT turbulence (Zhao et al. Reference Zhao, Wang, Liu and Lu2021), due to both occurring within cylindrical geometry. Given that the motions of fluid parcels are driven by buoyancy force and Coriolis force, the model equations for the inward and outward mixing layers in rotation-driven RT turbulence can be derived in sequence as

(3.7) \begin{align} \frac {{\textrm{d}}\dot {h}_{\textit{out}}}{{\textrm{d}}t} - \frac {\dot {h}_{\textit{out}}^2}{r_0+h_{\textit{out}}} &= 2\alpha _{\textit{out}} A_t \varOmega ^2(r_0+h_{\textit{out}}) - 2\varOmega \dot {h}_{\textit{out}}, \end{align}

(3.8) \begin{align} \frac {{\textrm{d}}\dot {h}_{\textit{in}}}{{\textrm{d}}t} + \frac {\dot {h}_{\textit{in}}^2}{r_0-h_{\textit{in}}} &= 2\alpha _{\textit{in}} A_t \varOmega ^2(r_0-h_{\textit{in}}) - 2\varOmega \dot {h}_{\textit{in}}. \end{align}

To obtain the analytical expression of the mixing widths, (3.7) and (3.8) are divided by $r_0+h_{\textit{out}}$ and $r_0-h_{\textit{in}}$ , respectively, allowing the left-hand sides of these equations to be formulated as

(3.9) \begin{align} \frac {1}{r_0+h_{\textit{out}}}\frac {{\textrm{d}}\dot {h}_{\textit{out}}}{{\textrm{d}}t} - \frac {\dot {h}_{\textit{out}}^2}{(r_0+h_{\textit{out}})^2} &= \frac {{\textrm{d}}}{{\textrm{d}}t}\left (\frac {\dot {h}_{\textit{out}}}{r_0+h_{\textit{out}}} \right)\!, \end{align}

(3.10) \begin{align} \frac {1}{r_0-h_{\textit{in}}}\frac {{\textrm{d}}\dot {h}_{\textit{in}}}{{\textrm{d}}t} + \frac {\dot {h}_{\textit{in}}^2}{(r_0-h_{\textit{in}})^2} &= \frac {{\textrm{d}}}{{\textrm{d}}t}\left (\frac {\dot {h}_{\textit{in}}}{r_0-h_{\textit{in}}} \right )\!. \end{align}

Therefore, after dividing (3.7) and (3.8) by $r_0+h_{\textit{out}}$ and $r_0-h_{\textit{in}}$ , respectively, these equations can be further formulated as

(3.11) \begin{equation} \frac {{\textrm{d}}}{{\textrm{d}}t}\left (\frac {\dot {h}_{\textit{out}}}{r_0+h_{\textit{out}}} \right ) = 2\alpha _{\textit{out}}A_t\varOmega ^2 -2 \varOmega \frac {\dot {h}_{\textit{out}}}{r_0+h_{\textit{out}}} \end{equation}

and

(3.12) \begin{equation} \frac {{\textrm{d}}}{{\textrm{d}}t}\left (\frac {\dot {h}_{\textit{in}}}{r_0-h_{\textit{in}}} \right ) = 2\alpha _{\textit{in}}A_t\varOmega ^2-2\varOmega \frac {\dot {h}_{\textit{in}}}{r_0-h_{\textit{in}}}, \end{equation}

respectively. Considering the initial mixing velocities $\dot {h}_{\textit{out}}(0)=0$ and $\dot {h}_{\textit{in}}(0)=0$ , and then integrating the above equations over $\dot {h}_{\textit{out}}/ ( r_0+h_{\textit{out}} )$ and $\dot {h}_{\textit{in}}/ ( r_0-h_{\textit{in}} )$ , gives

(3.13) \begin{equation} \frac {\dot {h}_{\textit{out}}}{r_0+h_{\textit{out}}} = \alpha _{\textit{out}} A_t \varOmega (1-{\textrm{e}}^{-2\varOmega t}) \end{equation}

and

(3.14) \begin{equation} \frac {\dot {h}_{\textit{in}}}{r_0-h_{\textit{in}}} = \alpha _{\textit{in}} A_t \varOmega (1-{\textrm{e}}^{-2\varOmega t}), \end{equation}

respectively. Finally, since the initial mixing widths are zero, the widths of the outward and inward mixing layers can be obtained by integrating (3.13) and (3.14), respectively, i.e.

(3.15) \begin{equation} \frac {h_{\textit{out}}}{r_0} = \exp \left [ \frac {1}{2}\alpha _{\textit{out}}A_t \left ({\textrm{e}}^{-2\varOmega t} + 2\varOmega t - 1 \right ) \right ]-1 \end{equation}

and

(3.16) \begin{equation} \frac {h_{\textit{in}}}{r_0} = 1 - \exp \left [ -\frac {1}{2}\alpha _{\textit{in}}A_t \left ({\textrm{e}}^{-2\varOmega t}+2\varOmega t -1 \right ) \right ]\!. \end{equation}

Based on the theoretical model of mixing width, the rescaled time $\tau =A_t ({\textrm{e}}^{-2\varOmega t}+ 2\varOmega t-1 )$ is a natural choice to describe the evolution of the mixing layer in rotation-driven RT turbulence with various $A_t$ and $\varOmega$ .

Despite the theoretical relations (3.15) and (3.16) having been derived, the dimensionless coefficients $\alpha _{\textit{out}}$ and $\alpha _{\textit{in}}$ still need to be determined to obtain usable models. In previous studies on planar RT turbulence, these coefficients are calculated using quantities related to mixing velocities and widths (Cook & Dimotakis Reference Cook and Dimotakis2001; Cabot & Cook Reference Cabot and Cook2006), i.e. $\alpha _i={\dot {h}^2} / 4 A_t g h$ . In a similar manner, the dimensionless coefficients for rotation-driven RT turbulence can be obtained based on (3.13) and (3.14) as

(3.17) \begin{equation} \alpha _{\textit{out}}= \frac {\dot h_{\textit{out}}(t)}{r_0+h_{\textit{out}}(t)} \frac {1}{A_t\varOmega (1-{\textrm{e}}^{-2\varOmega t})} \end{equation}

and

(3.18) \begin{equation} \alpha _{\textit{in}}= \frac {\dot h_{\textit{in}}(t)}{r_0-h_{\textit{in}}(t)} \frac {1}{A_t\varOmega (1-{\textrm{e}}^{-2\varOmega t})}, \end{equation}

respectively, which yield corresponding results shown in figure 17. It is evident that $\alpha _{\textit{out}}$ and $\alpha _{\textit{in}}$ tend to reach a quasi-steady value at $\tau \gtrsim 1$ for all cases listed in table 2. Therefore, both dimensionless coefficients $\alpha _{\textit{out}}$ and $\alpha _{\textit{in}}$ are approximately $0.3$ for rotation-driven RT turbulence.

Figure 17.

The dimensionless coefficients (a) $\alpha _{\textit{out}}$ and (b) $\alpha _{\textit{in}}$ in the theoretical models of mixing widths in rotation-driven RT turbulence.

Figure 18.

The positions of the outer and inner frontiers of the mixing layer in rotation-driven RT turbulence. Here, the lines represent the results of theoretical models, while the symbols denote the results of DNS.

To verify the accuracy of the theoretical models (3.15) and (3.16), the positions of the outward (i.e. $r_{\textit{out}}=r_0+h_{\textit{out}}$ ) and inward (i.e. $r_{\textit{in}}=r_0-h_{\textit{in}}$ ) frontiers of the mixing layer calculated by the models are compared with those obtained from the DNS results. As depicted in figure 18, the excellent agreement between the models and DNS results confirms that the current models (3.15) and (3.16) can be reliably used to predict the growth of the mixing layer in rotation-driven RT turbulence. Furthermore, the symbols of all cases collapse together well, indicating that the evolution of the mixing layer in rotation-driven RT turbulence can be unified based on the rescaled time $\tau$ .

It can be confirmed that the current models for rotation-driven RT turbulence can be reduced to quadratic growth for planar RT turbulence induced by constant external acceleration. To this end, the Taylor expansions of growth laws of mixing layers (3.15) and (3.16) are

(3.19) \begin{equation} h_{\textit{out}} = \alpha _{\textit{out}}A_t \varOmega ^2r_0 t^2 - \frac {2}{3}\alpha _{\textit{out}}A_t \varOmega ^3r_0 t^3 + \left (\frac {1}{2}\alpha _{\textit{out}}^2 A_t^2 + \frac {1}{3}\alpha _{\textit{out}}A_t \right ) \varOmega ^ 4r_0 t^4 + \cdots \end{equation}

and

(3.20) \begin{equation} h_{\textit{in}} = \alpha _{\textit{in}}A_t \varOmega ^2r_0 t^2 - \frac {2}{3}\alpha _{\textit{in}}A_t \varOmega ^3r_0 t^3 - \left ( \frac {1}{2}\alpha _{\textit{in}}^2 A_t^2 - \frac {1}{3}\alpha _{\textit{in}}A_t \right )\varOmega ^4r_0 t^4 + \cdots , \end{equation}

respectively. Moreover, the growth features of the mixing layer in rotation-driven RT turbulence manifest in the effects of geometrical curvature (i.e. $1/r_0$ ) and radial inhomogeneity of the centrifugal force (i.e. $\varOmega ^2(r_0\pm h_i$ ), where $+$ is for $h_{\textit{out}}$ , and $-$ is for $h_{\textit{in}}$ ) inducing the instability and Coriolis force ( $\varOmega \dot {h}_i$ ). However, these effects can be eliminated by taking limits with $\varOmega$ and $r_0$ at a fixed $A_t$ and a given time $t$ . Specifically, the effects of cylindrical geometry, governed by $1/r_0$ , becomes negligible as $r_0\to \infty$ . The effects of radial inhomogeneity of the centrifugal force, characterised by $\varOmega ^2(r_0\pm h_i)$ , and of the Coriolis force, determined by $2\varOmega \dot h_i$ , can be neglected as $\varOmega \to 0$ . It is worth noting that $\varOmega$ and $r_0$ should also satisfy the condition $\varOmega ^2 r_0 \rightarrow g$ with a constant $g$ to ensure the driving instability. As a result, the third- and higher-degree terms in the Taylor expansions (3.19) and (3.20) become infinitesimal at any given moment $t$ and fixed $A_t$ since $\varOmega ^mr_0 \rightarrow 0$ for $m\geqslant 3$ . Overall, (3.19) and (3.20) can be reduced as $h_{\textit{out}}=\alpha _{\textit{out}}A_t g t^2$ and $h_{\textit{in}}=\alpha _{\textit{in}}A_t g t^2$ . The reduced formulation of the mixing width is exactly the same as that in planar RT turbulence induced by constant external acceleration (Cook et al. Reference Cook, Cabot and Miller2004; Ristorcelli & Clark Reference Ristorcelli and Clark2004; Zhao et al. Reference Zhao, Liu and Lu2020a ). This analysis further mathematically confirms the reliability of the current theoretical models of mixing widths in rotation-driven RT turbulence.

Although the establishment of the model for rotation-driven RT turbulence follows the same approach as that for cylindrical RT turbulence (Zhao et al. Reference Zhao, Wang, Liu and Lu2021), their fundamentally different driving mechanisms preclude a direct reduction of the former model to the latter case. Indeed, there is a significant difference in the evolution of the mixing layer between the cylindrical RT turbulence and the rotation-driven RT turbulence, despite both cases sharing a similar flow configuration. It originates in the different driving mechanisms: cylindrical RT turbulence is driven by a constant external acceleration $g$ , whereas rotation-driven RT turbulence considered here is driven by rotation. Essentially, the evolution of the mixing layer in rotation-driven RT turbulence is governed by the combined effects of geometric curvature, the Coriolis force, and the radial inhomogeneity of the centrifugal force. Due to the fact that geometric curvature is nonlinearly coupled with the Coriolis force and the radial inhomogeneity of the centrifugal force, it is impossible to isolate the effect of geometric curvature while ignoring the other two factors. Specifically, as $\varOmega \rightarrow 0$ , the influences of the Coriolis force and the radial inhomogeneity of the centrifugal force become negligible. Meanwhile, to maintain the driving instability, $\varOmega$ and $r_0$ must satisfy the condition $\varOmega ^2 r_0\rightarrow g$ with a constant $g$ . This implies that $r_0 \sim \varOmega ^{-2} \rightarrow \infty$ , resulting in the geometric curvature’s impact becoming negligible in this regime. Therefore, neither factor can be disregarded independently, preventing a direct reduction of the centrifugal RT turbulence model to the cylindrical RT turbulence model (Zhao et al. Reference Zhao, Wang, Liu and Lu2021).