2.1. Governing equations and numerical method

In the current simulations, non-reactive mono-species gases and Newtonian fluids are considered, for which the dynamic viscosity relies only on the temperature. The dimensionless governing equations for compressible turbulence in vibrational non-equilibrium can be written as follows (Donzis & Maqui Reference Donzis and Maqui2016; Zheng et al. Reference Zheng, Wang, Alam, Noack, Li, Wan and Chen2021):

(2.1)\begin{gather} \frac{\partial {\rho}}{\partial {t}} + \frac{\partial(\rho u_j)}{\partial x_j} = 0 \end{gather}

(2.2)\begin{gather}\frac{\partial{(\rho u_i)}}{\partial {t}} + \frac{\partial{[\rho u_i u_j + p {\boldsymbol{\delta}}_{ij}]}}{\partial{x_j}} = \frac{1}{Re} \frac{\partial{\sigma_{ij}}}{\partial{x_j}} + \mathcal{F}_i \end{gather}

(2.3)\begin{gather}\frac{\partial{\varepsilon}}{\partial{t}} + \frac{\partial{[(\varepsilon + p)u_j]}}{\partial{x_j}}= \frac{1}{\alpha} \frac{\partial}{\partial{x_j}}\left(\kappa_{tr} \frac{\partial{T_{tr}}}{\partial{x_j}} + \kappa_v \frac{\partial{T_v}}{\partial{x_j}} \right) + \frac{1}{Re}\frac{\partial{(\sigma_{ij}u_i)}}{\partial{x_j}} - \varLambda + \mathcal{F}_I + \mathcal{F}_ju_j \end{gather}

(2.4)\begin{gather}\frac{\partial{E_v}}{\partial{t}} + \frac{\partial{(E_vu_j)}}{\partial{x_j}} = \frac{1}{\alpha} \frac{\partial}{\partial{x_j}}\left(\kappa_v \frac{\partial{T_v}}{\partial{x_j}}\right) + \frac{E_v^*-E_v}{\tau_v} \end{gather}

(2.5)\begin{gather}p = \rho T_{tr}/(\gamma_r M^2) \end{gather}

where $\rho$, $u_i$, $p$, $T_{tr}$ and $T_v$ are the dimensionless density, velocity components, pressure, translational–rotational and vibrational temperatures, respectively. The total energy $\varepsilon$ includes the kinetic energy ($\rho u_j u_j /2$), and the translational–rotational ($E_{tr} = 5\rho T_{tr}/(2\gamma _r M^2)$) and vibrational ($E_v$) energies. The dimensionless, large-scale forcings to the fluid momentum and the translational–rotational energy are respectively denoted as $\mathcal {F}_j$ and $\mathcal {F}_I$, which are explained in more detail in the appendix of Zheng et al. (Reference Zheng, Wang, Alam, Noack, Li, Wan and Chen2021). A spatially uniform thermal cooling function $\varLambda$ is adopted to sustain the internal energy in a statistically steady state. In the equation of state (2.5), only the translational–rotational temperature $T_{tr}$ is adopted since the pressure mainly stems from the translational motion of molecules rather than the rotational and vibrational motions (Vincenti & Kruger Reference Vincenti and Kruger1965). The reference Reynolds number $Re = \rho _r U_r L_r/\mu _r$, the reference Mach number $M = U_r/c_r$ and the reference Prandtl number $Pr = \mu _r C_{p_r}/\kappa _r$ are three governing parameters. Here, $\rho _r$, $U_r$, $L_r$ and $\mu _r$ are respectively the reference density, velocity, length and viscosity coefficient. The reference speed of sound is given by $c_r = \sqrt {\gamma _r R T_r}$, where $R$ is the specific gas constant and $T_r = 1200$ K is the reference temperature. The parameter $\gamma _r = C_{p_r}/C_{v_r}$ is the ratio of specific heat at constant pressure $C_{p_r}$ to that at constant volume $C_{v_r}$, approximately equalling 1.324 according to the ratio of specific heats of dry air at $T_r = 1200$ K. The dimensionless parameter $\alpha$ is defined as $\alpha \equiv PrRe(\gamma _r-1)M^2$, where $Pr$ equals 0.71.

The vibrational energy per unit volume in equilibrium ($E_v^*$) and non-equilibrium ($E_v$) for diatomic molecules, and the viscosity stress $\sigma _{ij}$ are given as

(2.6)\begin{gather} E_{v}^* = \frac{\rho \theta_{v}}{\gamma_r M^2 [\exp(\theta_{v}/T_{tr}) -1]}, \end{gather}

(2.7)\begin{gather}E_{v} = \frac{\rho \theta_{v}}{\gamma_r M^2 [\exp(\theta_{v}/T_v) -1]}, \end{gather}

and

(2.8)\begin{equation} \sigma_{ij} = \mu \left(\frac{\partial{u_i}}{\partial{x_j}} + \frac{\partial{u_j}}{\partial{x_i}}\right) - \frac{2}{3}\mu\theta{\boldsymbol{\delta}}_{ij}. \end{equation}

The parameter $\theta _v$ is the characteristic vibrational temperature normalized by $T_r$, while $\theta = \partial u_k/\partial x_k$ is the velocity divergence. The temperature-dependent viscosity ($\mu$) and thermal conductivity coefficients ($\kappa _{tr}$ and $\kappa _{v}$) are specified by the Sutherland and Eucken laws (Vincenti & Kruger Reference Vincenti and Kruger1965; Anderson Reference Anderson2006). For their detailed expressions, please refer to our previous publication (Zheng et al. Reference Zheng, Wang, Noack, Li, Wan and Chen2020).

The vibrational rate $Q_v = (E_v^*-E_v)/\tau _v$ in the vibrational energy governing equation (2.4) is based on the widely used Landau–Teller relaxation model. The dimensionless relaxation time ($\tau _v$) relies closely on the local temperature and pressure, and is roughly calculated by

(2.9)\begin{equation} \tau_v = (C/p)\exp(K_2/T_{tr})^{1/3}. \end{equation}

Here, $C$ and $K_2$ are dimensionless constants relating to the molecular structure of gases. In the current simulations, the dimensionless parameter $\langle K_\tau \rangle = \langle \tau _v \rangle /\tau _\eta$ is adopted to characterize the time scale of the relaxation process, where the $\langle\, {\cdot }\, \rangle$ operator stands for the spatial average. Also, $\tau _\eta = [\langle \mu /(Re\rho )\rangle /\epsilon ]^{1/2}$ is the Kolmogorov time scale, and $\epsilon =\langle \sigma _{ij}{\mathsf{S}}_{ij}/Re \rangle / \langle \rho \rangle$ is the kinetic energy dissipation rate due to viscosity. The value of $K_2$ is set to be 2000.0, while the constant $C$ is adjusted to obtain a specific $\langle K_\tau \rangle$ value.

The governing equations of compressible turbulence are numerically solved in a cubic box with a side length equalling $2{\rm \pi}$ and a $512^3$ grid resolution. Periodic boundary conditions are adopted in all three spatial directions. The hybrid compact-weighted essentially non-oscillatory (compact-WENO) scheme is applied, which couples a eighth-order central compact finite difference scheme in smooth regions with a seventh-order WENO scheme in shock regions (Lele Reference Lele1992; Balsara & Shu Reference Balsara and Shu2000; Wang et al. Reference Wang, Wang, Xiao, Shi and Chen2010). The time derivative is approximated by the total variation diminishing (TVD) Runge–Kutta method (Gottlieb & Shu Reference Gottlieb and Shu1998). Following Samtaney, Pullin & Kosović (Reference Samtaney, Pullin and Kosović2001), the velocity field, which is divergence free, is initialized using a random field with a specified spectrum ($E(k) = 0.011 k^4 \exp (-k^2/8)$, where $k$ is the wavenumber). Meanwhile, the normalized temperatures ($T_{tr}$ and $T_v$) and density ($\rho$) are initialized with constant values (1.0) at all spatial points, and the initial pressure is determined from the equation of state (2.5). After the system reaches the statistically stationary state, 61 flow fields, uniformly spanning the time period of $9.01 \lesssim t/T_e \lesssim 14.41$, are adopted to obtain the statistical averages of quantities. Here, $T_e (= \sqrt {3}L_f/u^\prime )$ is the large eddy turnover time and $L_f$ is the integral length scale.

2.2. Forcing strategy

The velocity field ${\boldsymbol {u}}({\boldsymbol {x}},t)$ is transformed into the wave space using the Fourier transform, and further decomposed into a solenoidal field ($\hat {{\boldsymbol {u}}}^S({\boldsymbol {k}},t)$) and a dilatational field ($\hat {{\boldsymbol {u}}}^D({\boldsymbol {k}},t)$) based on the Helmholtz decomposition, where ${\boldsymbol {k}}$ is the wave vector. The kinetic energy per unit mass for each wave vector is thus decomposed as follows:

(2.10)\begin{equation} \frac{|\hat{{\boldsymbol{u}}}({\boldsymbol{k}},t)|^2}{2}= \frac{|\hat{{\boldsymbol{u}}}^S({\boldsymbol{k}},t)|^2}{2}+ \frac{|\hat{{\boldsymbol{u}}}^D({\boldsymbol{k}},t)|^2}{2}. \end{equation}

The kinetic energy in each of the first two wavenumber shells is calculated as

(2.11)\begin{equation} E^u(0.5 \leq k < 1.5) = \sum_{0.5 \leq | {\boldsymbol{k}} | < 1.5} \left(\frac{|\hat{{\boldsymbol{u}}}({\boldsymbol{k}},t)|^2}{2}\right), \end{equation}

and

(2.12)\begin{equation} E^u(1.5 \leq k < 2.5) = \sum_{1.5 \leq | {\boldsymbol{k}} | < 2.5}\left(\frac{|\hat{{\boldsymbol{u}}}({\boldsymbol{k}},t)|^2}{2}\right). \end{equation}

Similarly, the kinetic energy in the first two wavenumber shells is decomposed as

(2.13)\begin{equation} E^u(0.5 \leq k < 1.5) = E^{u,S}(0.5 \leq k < 1.5)+E^{u,D}(0.5 \leq k < 1.5), \end{equation}

and

(2.14)\begin{equation} E^u(1.5 \leq k < 2.5) = E^{u,S}(1.5\leq k < 2.5)+E^{u,D}(1.5\leq k < 2.5). \end{equation}

The large-scale momentum forcing is applied to the solenoidal velocity component, while the dilatational velocity component is left untouched (Petersen & Livescu Reference Petersen and Livescu2010; Wang et al. Reference Wang, Wang, Xiao, Shi and Chen2010; Donzis & Jagannathan Reference Donzis and Jagannathan2013). To maintain the total kinetic energy in the first two shells to the prescribed levels $E_u(1)$ and $E_u(2)$, respectively, the solenoidal velocity component is amplified. The forced velocity field $\hat {{\boldsymbol {u}}}^f({\boldsymbol {k}},t)$ is given as

(2.15)\begin{equation} \hat{{\boldsymbol{u}}}^f({\boldsymbol{k}},t)=\alpha \hat{{\boldsymbol{u}}}^S({\boldsymbol{k}},t)+\hat{{\boldsymbol{u}}}^D({\boldsymbol{k}},t), \end{equation}

where $\alpha$ for all modes in each wavenumber shell is set to be

(2.16)\begin{equation} \alpha(0.5 \leq k < 1.5) = \sqrt{\frac{E^u(1)-E^{u,D}(0.5 \leq k < 1.5)}{E^{u,S}(0.5 \leq k < 1.5)}}, \end{equation}

and

(2.17)\begin{equation} \alpha(1.5 \leq k < 2.5) = \sqrt{\frac{E^u(2)-E^{u,D}(1.5\leq k < 2.5)}{E^{u,S}(1.5\leq k < 2.5)}}, \end{equation}

where $E^u(1) = 1.242$ and $E^u(2) = 0.391$.

The large-scale thermal forcing for the translational–rotational temperature field is similar to that for the solenoidal velocity field (Donzis & Maqui Reference Donzis and Maqui2016; Wang et al. Reference Wang, Wan, Chen, Xie, Wang and Chen2019). The translational–rotational temperature field $T_{tr}({\boldsymbol {x}},t)$ is transformed into Fourier space to yield $\hat {T}_{tr}({\boldsymbol {k}},t)$. Similarly,

(2.18)\begin{equation} E^{T_{tr}}(0.5 \leq k < 1.5) = \sum_{0.5 \leq | {\boldsymbol{k}} | < 1.5} (|\hat{T}_{tr}({\boldsymbol{k}},t)|^2), \end{equation}

and

(2.19)\begin{equation} E^{T_{tr}}(1.5 \leq k < 2.5) = \sum_{1.5 \leq | {\boldsymbol{k}} | < 2.5} (|\hat{T}_{tr}({\boldsymbol{k}},t)|^2). \end{equation}

The forced translational–rotational temperature is given as

(2.20)\begin{equation} \hat{T}_{tr}^f({\boldsymbol{k}},t) = \beta \hat{T}_{tr}({\boldsymbol{k}},t), \end{equation}

where $\beta$ for all modes in each wavenumber shell is set to be

(2.21)\begin{equation} \beta(0.5 \leq k < 1.5) = \sqrt{\frac{E^{T_{tr}}(1)}{{E^{T_{tr}}(0.5 \leq k < 1.5)}}}, \end{equation}

and

(2.22)\begin{equation} \beta(1.5 \leq k < 2.5) = \sqrt{\frac{E^{T_{tr}}(2)}{{E^{T_{tr}}(1.5 \leq k < 2.5)}}}, \end{equation}

where $E^{T_{tr}}(1) = E^u(1)/100$ and $E^{T_{tr}}(2) = E^u(2)/100$.

In present simulations, the momentum and thermal forcings act on the large scales, and are expected to have a small impact on the statistics properties in the inertial regime. However, the large-scale thermal forcing in the present simulations cannot completely reproduce the shock-induced heating in the high-speed flows of practical interest.

4.1. Dilatational effect on enstrophy production

From the vorticity transport equation, the governing equation for enstrophy ($\omega ^2/2$) can be derived as follows (Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2011; Papapostolou et al. Reference Papapostolou, Wacks, Chakraborty, Klein and Im2017):

(4.1)\begin{align} \left(\frac{\partial}{\partial t}+u_j\frac{\partial}{\partial x_j} \right) \frac{\omega^2}{2} &= \omega _i \omega _j {\mathsf{S}}_{ij}- \omega ^2\theta + \omega _i\frac{\epsilon _{ijk}}{\rho ^2}\frac{\partial \rho}{\partial x_j} \frac{\partial p}{\partial x_k} \nonumber\\ &\quad +\omega _i \frac{\epsilon _{ijk}}{Re} \frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \sigma_{mk}}{\partial x_m} \right) + \omega_i \epsilon _{ijk} \frac{\partial \mathcal{F}_k}{\partial x_j}. \end{align}

In (4.1), $P_\omega = \omega _i \omega _j {\mathsf{S}}_{ij}$ is the vortex stretching and tilting (strain) term, $D_\omega = - \omega ^2\theta$ is the dilatational term, $B_\omega = (1/\rho ^2)\varepsilon _{ijk}\omega _i(\partial \rho / \partial x_j)(\partial p / \partial x_j)$ is the baroclinic term, $V_\omega = (1/Re)\omega _i \epsilon _{ijk}(\partial /\partial x_j)(\partial \sigma / \partial x_m)$ is the viscous term and $F_\omega = \omega _i \epsilon _{ijk}(\partial \mathcal {F}_k / \partial x_j)$ is the large-scale forcing term. The parameter $F_\omega$ is expected to have a small impact on the local enstrophy production because the momentum forcing is at large scales. The local enstrophy production thus mainly includes contributions from the strain, dilatational and baroclinic effects. However, the contribution from the baroclinic term is negligible in compressible isotropic turbulence (Wang et al. Reference Wang, Shi, Wang, Xiao, He and Chen2011). The specific focus in the current analysis is therefore on the strain and dilatational terms.

Equation (4.1) can be further derived as

(4.2)\begin{align} \frac{\partial (\omega ^2/2)}{\partial t}+\frac{\partial (u_j \omega ^2/2)}{\partial x_j}= \omega _i\omega _j {\mathsf{S}}^*_{ij}+\omega _i\frac{\epsilon _{ijk}}{\rho ^2}\frac{\partial \rho}{\partial x_j} \frac{\partial p}{\partial x_k}+\omega _i\frac{\epsilon _{ijk}}{Re}\frac{\partial}{\partial x_j} \left(\frac{1}{\rho}\frac{\partial \sigma _{mk}}{\partial x_m} \right) + \omega _i \epsilon _{ijk} \frac{\partial \mathcal{F}_k}{\partial x_j}, \end{align}

where the component of the modified strain-rate tensor ${\mathsf{S}}^*_{ij}={\mathsf{S}}_{ij}-{\mathsf{S}}_{kk}{\boldsymbol{\delta}}_{ij}/2={\mathsf{D}}_{ij}-{\mathsf{S}}_{kk}{\boldsymbol{\delta}}_{ij}/6$. The net enstrophy production term ($\omega _i\omega _j {\mathsf{S}}^*_{ij}$) thus can be rewritten as

(4.3)\begin{equation} \omega_i\omega _j {\mathsf{S}}^*_{ij} = \omega_i\omega _j {\mathsf{D}}^S_{ij} + \omega_i\omega _j {\mathsf{D}}^D_{ij} - \frac{1}{6}\theta \omega ^2. \end{equation}

The spatial averages of the net enstrophy production and its components for the $M_t \approx 0.22$ and $0.68$ cases are listed in table 7. It is clearly observed that $\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle \approx \langle \omega _i\omega _j {\mathsf{D}}^S_{ij} \rangle$, while $\langle \omega _i\omega _j {\mathsf{D}}^D_{ij} \rangle$ and $\langle -(1/6)\theta \omega ^2 \rangle$ are negligible. This indicates that, from the full flow field perspective, the enstrophy production mainly stems from the solenoidal component of the deviatoric strain-rate tensor. However, there is no obvious relationship between the spatial averages of the net enstrophy production (or its components) and the vibrational relaxation. Figure 4 illustrates the instantaneous contours of $\omega /\omega ^\prime$, $\omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, $\omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $-(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ for case $\mathrm {I}_5$. While the correlation coefficients between vorticity and different enstrophy production terms (including their absolute values) are listed in table 8. It is found that the structures of the vorticity ($\omega /\omega ^\prime$) and enstrophy production terms ($\omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, $\omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $-(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$) are similar, for example, as marked by the circles with a dashed line. Consequently, the vorticity has a strong correlation with the absolute values of the enstrophy production terms. The correlation coefficients are larger than 0.53 (table 8). On the other hand, unlike the global averages in table 7, the local magnitudes of $\omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $-(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ can be considerable, especially for cases with a weak relaxation effect (e.g. case $\mathrm {I}_5$, figure 4c,d). However, the positive values in the compression region and the negative values in the expansion region cancel out on average, leading to a negligible amount. This is why $\textrm {Corr}(\omega /\omega ^\prime, \mathcal {W}({\mathsf{D}}^D_{ij}))$ and $\textrm {Corr}(\omega /\omega ^\prime, \mathcal {W}(-\theta /6))$ are close to zero (table 8).

Figure 4. Instantaneous contours of (a) $\omega /\omega ^\prime$, (b) $\omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, (c) $\omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, (d) $-(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$. Here, x = 3.14 and $\omega ^\prime = \sqrt {\langle \omega _1^2 + \omega _2^2 + \omega _3^2 \rangle }$. Case $\mathrm {I}_5$.

Table 7. Spatial averages of the net enstrophy production and its components; $\mathcal {F}({\mathsf{S}}^*_{ij}) = \langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle$, $\mathcal {F}({\mathsf{D}}^S_{ij}) = \langle \omega _i \omega _j {\mathsf{D}}^S_{ij} \rangle$, $\mathcal {F}({\mathsf{D}}^D_{ij}) = \langle \omega _i \omega _j {\mathsf{D}}^D_{ij} \rangle$ and $\mathcal {F}(-\theta /6) = \langle -\omega ^2\theta /6 \rangle$.

Table 8. Correlation coefficients between vorticity and enstrophy production terms (including their absolute values). Case $\mathrm {I}_5$; $\mathcal {W}({\mathsf{S}}^*_{ij}) = \omega _i \omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle$, $\mathcal {W}({\mathsf{D}}^S_{ij}) = \omega _i \omega _j {\mathsf{D}}^S_{ij}/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle$, $\mathcal {W}({\mathsf{D}}^D_{ij}) = \omega _i \omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle$ and $\mathcal {W}(-\theta /6) = (-\omega ^2\theta /6) /\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle$.

Figures 5 and 6 present the spatial averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$, $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$, $\omega _i\omega _j {\mathsf{D}}^S_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ and $-(1/6)\theta \omega ^2/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ conditioned on the local dilatation for the $M_t \approx 0.22$ and $0.68$ cases, respectively. As illustrated in figure 5(a), the conditional averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ among the $M_t \approx 0.22$ cases are very different from each other. The conditional averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ approximately equal 1.0 at $\theta /\theta ^\prime = 0.0$. When the relaxation effect is significant (e.g. cases $\mathrm {I}_1$– $\mathrm {I}_3$), the conditional averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ increase in the strong expansion and compression regions, and case $\mathrm {I}_3$ exhibits the largest growth rate. For cases with a weak relaxation effect (e.g. cases $\mathrm {I}_4$– $\mathrm {I}_7$), the conditional averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ increase slightly in the compression region, and gradually change from increasing to decreasing in the expansion region as the relaxation effect fades. For instance, the conditional average of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ for case $\mathrm {I}_5$ decreases from 1.0 to 0.0 in the range of $0.0 \leq \theta /\theta ^\prime \leq 4.0$, and is negative at $\theta /\theta ^\prime > 4.0$. In figure 5(b), the conditional averages of $\omega _i\omega _j {\mathsf{D}}^S_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ for different cases are positive in both the compression and expansion regions. Their variations are similar to those of $\omega ^2/\langle \omega ^2 \rangle$ and ${\mathsf{D}}^S_{ij}{\mathsf{D}}^S_{ij}/\langle {\mathsf{D}}^S_{ij}{\mathsf{D}}^S_{ij}\rangle$ (figure 3a,b). Furthermore, the magnitudes of $\omega _i\omega _j {\mathsf{D}}^S_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ are comparable to those of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$, which agrees with the observations in table 7. For $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ and $-(1/6)\theta \omega ^2/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$, their magnitudes are much smaller than those of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$. The magnitudes of $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ for different cases roughly equal 0.0 in the range of $-2.0 \leq \theta /\theta ^\prime \leq 2.0$. Beyond this range, the magnitudes of $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ increase with the local dilatation in both compression and expansion regions, and their growth rates are enhanced as the relaxation effect increases (figure 5c). The conditional averages of $-(1/6)\theta \omega ^2/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ for different cases are positive in the compression region, and negative in the expansion region; their magnitudes increase with the local dilatation (figure 5d). Interestingly, their growth rates are weakened as the relaxation effect increases, different from the $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ term.

Figure 5. Conditional averages of the net enstrophy production and its components; (a) $\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$, (b) $\langle \omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$, (c) $\langle \omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$ and (d) $\langle -(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$. Here, $M_t \approx 0.22$.

Figure 6. Conditional averages of the net enstrophy production and its components; (a) $\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$, (b) $\langle \omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$, (c) $\langle \omega _i \omega _j {\mathsf{D}}_{ij}^{D} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$ and (d) $\langle -(1/6)\omega ^2\theta /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle \mid \theta /\theta ^\prime \rangle$. Here, $M_t \approx 0.68$.

For the $M_t \approx 0.68$ cases (figure 6), the conditional averages of net enstrophy production and its components are closely related to the local dilatation and vibrational relaxation. The conditional averages of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ for different cases almost increase linearly with the local dilatation in the compression region, and their growth rates are weakened by the relaxation effect (figure 6a). In the expansion region, behaviours of $\omega _i\omega _j {\mathsf{S}}^*_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ are rather complicated: they first increase with the local dilatation, then decrease from positive to negative. The weaker the relaxation effect, the sharper the decrease. This suggests that the compression enhances the enstrophy production; however, the high expansion weakens it. In the extreme expansion region (e.g. $\theta /\theta ^\prime > 4.0$ for case $\mathrm {II}_5$), $\omega _i\omega _j {\mathsf{S}}^*_{ij}$ can even become a enstrophy destruction term. As shown in figure 6(b), the conditional averages of $\omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ for different cases are nearly not affected by the local dilatation, remaining almost constant in the compression region. In the expansion region, the conditional averages of $\omega _i \omega _j {\mathsf{D}}_{ij}^{S} /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are enlarged. Their differences among different cases are mainly manifested in the strong expansion region. Variations of $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ and $-(1/6)\theta \omega ^2/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ with the local dilatation are similar (figure 6c,d). They are positive in the compression region and negative in the expansion region, and their magnitudes increase with the local dilatation. Meanwhile, their growth rates are weakened by the relaxation effect in both compression and expansion regions. However, the magnitude of $\omega _i\omega _j {\mathsf{D}}^D_{ij}/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ is smaller than that of $-(1/6)\theta \omega ^2/\langle \omega _i\omega _j {\mathsf{S}}^*_{ij} \rangle$ with the same dilatational level.

4.2. Dilatational effect on deviatoric strain-rate tensor

The interaction between strain and vorticity can be described in the eigenframe of the (deviatoric) strain-rate tensor. The modified strain-rate tensor is diagonalized into an orthonormal basis, with its eigenvalues given by $\lambda _i$ for $i = 1, 2, 3$ (with $\lambda _1 \leq \lambda _2 \leq \lambda _3$) and the corresponding eigenvectors denoted by $\boldsymbol {\varLambda }_{\boldsymbol {i}}$. Consequently, the net enstrophy production term can be rewritten as

(4.4)\begin{equation} \omega_i\omega_j {\mathsf{S}}^*_{ij} = \omega^2 [ \lambda_1 \cos^2(\omega,\varLambda_1)+ \lambda_2 \cos^2(\omega,\varLambda_2)+\lambda_3 \cos^2(\omega,\varLambda_3)], \end{equation}

which isolates the individual contribution from each eigendirection. Similarly, the enstrophy production terms due to the solenoidal and dilatational components of the deviatoric strain-rate tensor ($\omega _i\omega _j {\mathsf{D}}^{S}_{ij}$ and $\omega _i\omega _j {\mathsf{D}}^{D}_{ij}$) can be rewritten as

(4.5)\begin{equation} \omega_i\omega_j {\mathsf{D}}^{S}_{ij} = \omega^2 [\lambda^{S}_1 \cos^2(\omega,\varLambda^{S}_1)+\lambda^{S}_2 \cos^2(\omega,\varLambda^{S}_2)+\lambda^{S}_3 \cos^2(\omega,\varLambda^{S}_3)], \end{equation}

and

(4.6)\begin{equation} \omega_i\omega_j {\mathsf{D}}^{D}_{ij} = \omega^2 [ \lambda^{D}_1 \cos^2(\omega,\varLambda^{D}_1)+\lambda^{D}_2 \cos^2(\omega,\varLambda^{D}_2)+\lambda^{D}_3 \cos^2(\omega,\varLambda^{D}_3)]. \end{equation}

Here, $\lambda ^{S}_i$ (or $\lambda^D_i$) and $\varLambda ^{S}_i$ (or $\varLambda^D_i$) are the eigenvalues and eigenvectors of the solenoidal (or dilatational) component of the deviatoric strain-rate tensor ($\boldsymbol{\mathsf{D}}$).

The first eigenvalue and the third eigenvalue always make a negative and a positive contribution to the enstrophy production, respectively, while the contribution of the second eigenvalue relies on its sign and magnitude. In addition, the orientation of the vorticity relative to the strain-rate eigenvector also plays a significant role in the enstrophy production. In the following discussions, effects of the vibrational relaxation and local dilatation on the statistical properties of strain-rate eigenvalues, alignments between the vorticity and strain-rate eigenvectors and individual contributions to enstrophy production from each eigendirection will be examined step by step.

In tables 9–12, the spatial averages of the eigenvalues, alignments between the vorticity and strain eigenvectors and the individual contributions to the enstrophy production in each eigendirection are summarized. The variable $\beta _i^X$ ($X$ denotes $D$ or $S$) is defined as

(4.7)\begin{equation} \beta_i^X = \frac{\lambda_i^X}{\sqrt{(\lambda_1^X)^2+(\lambda_2^X)^2+(\lambda_3^X)^2}}, \end{equation}

which represents the relative amplitude of $\lambda _i^X$. For the solenoidal component of the deviatoric strain-rate tensor (tables 9 and 11), the vibrational relaxation has little influence on the ratio of $\langle \beta ^S_i \rangle$. The ratio of $\langle \beta ^S_i \rangle$ is approximately $-$0.743:0.117:0.626 for both the $M_t \approx 0.22$ and $0.68$ cases. The ratio of $\langle \textrm {Cos}^2(\omega, \varLambda ^S_i) \rangle$ roughly equals 0.158:0.528:0.314 for the $M_t \approx 0.22$ cases, and 0.155:0.542:0.303 for the $M_t \approx 0.68$ cases. The alignment between the vorticity and the third eigenvector is enhanced slightly for the $M_t \approx 0.22$ cases, such that the dominant contribution to $\omega _i \omega _j{\mathsf{D}}_{ij}^S$ arises from the third eigendirection for the $M_t \approx 0.22$ cases, while it results from the second eigendirection for the $M_t \approx 0.68$ cases (tables 9 and 11). The ratio of $\langle \omega ^2 \lambda _i^S\textrm {Cos}^2(\omega, \varLambda ^S_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ roughly equals $-$0.348:0.632:0.716 and $-$0.285:0.672:0.619 for the $M_t \approx 0.22$ and $0.68$ cases, respectively. Here, the sum of $\langle \omega ^2 \lambda _i^S\textrm {Cos}^2(\omega, \varLambda ^S_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ for each case is approximately 1.0, implying that the net enstrophy production mainly stems from the solenoidal component of the deviatoric strain-rate tensor from the full flow field perspective. This observation agrees with the results in table 7.

Table 9. Statistics of various quantities in three eigendirections about $\omega _i \omega _j {\mathsf{D}}^S_{ij}$. Here, $M_t \approx 0.22$.

Table 10. Statistics of various quantities in three eigendirections about $\omega _i \omega _j {\mathsf{D}}^D_{ij}$. Here, $M_t \approx 0.22$.

Table 11. Statistics of various quantities in three eigendirections about $\omega _i \omega _j {\mathsf{D}}^S_{ij}$. Here, $M_t \approx 0.68$.

Table 12. Statistics of various quantities in three eigendirections about $\omega _i \omega _j {\mathsf{D}}^D_{ij}$. Here, $M_t \approx 0.68$.

Values of $\langle \beta ^D_i \rangle$, $\langle \textrm {Cos}^2(\omega, \varLambda ^D_i) \rangle$ and $\langle \omega ^2\lambda _i^D\textrm {Cos}^2(\omega, \varLambda ^D_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ (tables 10 and 12) are very different from those for the solenoidal component of the deviatoric strain-rate tensor. The magnitudes of $\langle \beta ^D_1 \rangle$ and $\langle \beta ^D_3 \rangle$ are close to each other, while that of $\langle \beta ^D_2 \rangle$ is much smaller. In addition, $\langle \beta ^D_2 \rangle$ and $\langle \beta ^D_3 \rangle$ are positive, and $\langle \beta ^D_1 \rangle + \langle \beta ^D_2 \rangle + \langle \beta ^D_3 \rangle = 0.0$. The ratio of $\langle \textrm {Cos}^2(\omega, \varLambda ^D_i) \rangle$ approximately equals 0.326:0.347:0.327 for the $M_t \approx 0.22$ cases, and 0.318:0.370:0.311 for the $M_t \approx 0.68$ cases. The value of $\langle \textrm {Cos}^2(\omega, \varLambda ^D_i) \rangle$ roughly approaches $1/3$ for each eigenvector, implying that $\omega$ and $\varLambda ^D_i$ are nearly distributed independently and uniformly from the full flow field perspective. The sum of $\langle \omega ^2 \lambda _i^D\textrm {Cos}^2(\omega, \varLambda ^D_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ for each case is approximately 0.0, indicating the negligible contribution from the dilatational component of the deviatoric strain-rate tensor to the net enstrophy production. The magnitudes of $\langle \omega ^2 \lambda _1^D\textrm {Cos}^2(\omega, \varLambda ^D_1)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ and $\langle \omega ^2 \lambda _3^D\textrm {Cos}^2(\omega, \varLambda ^D_3)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ are comparable, while that of $\langle \omega ^2 \lambda _2^D\textrm {Cos}^2(\omega, \varLambda ^D_2)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ is one order of magnitude smaller than the other two components. This should be attributed to the smaller $\langle \beta ^D_2 \rangle$. Meanwhile, the magnitude of $\langle \omega ^2 \lambda _i^D\textrm {Cos}^2(\omega, \varLambda ^D_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ is enlarged as the relaxation effect fades. For instance, the ratios of $\langle \omega ^2 \lambda _i^D\textrm {Cos}^2(\omega, \varLambda ^D_i)/\langle \omega _i \omega _j {\mathsf{S}}^*_{ij} \rangle \rangle$ are respectively approximately $-$0.036:0.001:0.033 and $-$0.224:0.022:0.204 for cases $\textrm {I}_3$ and $\textrm {I}_5$ (table 10).

Figure 7 shows the PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^S$ for cases $\textrm {I}_3$ and $\textrm {I}_5$, and figure 8 for cases $\textrm {II}_3$ and $\textrm {II}_5$. The three normalized eigenvalues of ${\mathsf{D}}_{ij}^S$ are respectively located in the ranges of $[-0.816, -0.408]$, $[-0.408, 0.408]$ and $[0.408, 0.816]$. For cases $\textrm {I}_3$ and $\textrm {II}_3$, where the relaxation effect is significant, the lines corresponding to different dilatational levels have a slight deviation; however, their peaks are close (figures 7(a), 8(a)). When the relaxation effect declines (e.g. cases $\textrm {I}_5$ and $\textrm {II}_5$), the lines corresponding to different dilatational levels almost overlap each other (figures 7(b), 8(b)). The effects of local dilatation and vibrational relaxation on the PDF of $\beta _i^S$ are negligible, and this is consistent with the observations in tables 9 and 11. In the overall flow field, the preferred eigenvalue ratio is approximately $-$4.0:1.0:3.0, in good agreement with the results in incompressible turbulent flows (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987).

Figure 7. The PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^S$. (a) Case $\textrm {I}_3$ and (b) case $\textrm {I}_5$.

Figure 8. The PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^S$. (a) Case $\textrm {II}_3$ and (b) case $\textrm {II}_5$.

Figures 9 and 10 present the PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^D$ for cases $\textrm {I}_3$ and $\textrm {I}_5$, as well as cases $\textrm {II}_3$ and $\textrm {II}_5$, respectively. Although the conditional PDFs of $\beta _i^D$ are closely related to the local dilatational and vibrational relaxation, the preferred ratio of $\beta _i^D$ is approximately $-$0.81:0.1:0.80 in the overall flow field for both the $M_t \approx 0.22$ and $0.68$ cases. The first eigenvalue with $\beta _1^D \approx -0.81$ dominates in compression regions, while the third eigenvalue with $\beta _3^D \approx 0.80$ dominates in expansion regions (figures 9(a,c,d,f) and 10(a,c,d,f)). When the flow compressibility is weak (e.g. case $\textrm {I}_3$), the peak value of the conditional PDF of $\beta _1^D$ in the compression region is much smaller than the other cases. Furthermore, the conditional PDFs of $\beta _1^D$ do not have an obvious peak in the expansion region. The intermediate eigenvalue $\beta _2^D$ tends to be positive in the compression region, and negative in the expansion region. As the flow compressibility increases, this tendency becomes obvious (figures 9(b,e) and 10(b,e)). The third eigenvalue $\beta _3^D$ is positive. The value of $\beta _3^D$ in the compression region becomes smaller with the increase of flow compressibility, while that in the expansion region almost does not change. For instance, the preferred $\beta _2^D$ and $\beta _3^D$ in the strong compression region approximately equal 0.11 and 0.79, respectively, for case $\textrm {I}_3$; these values roughly equal 0.34 and 0.47, respectively, for case $\textrm {I}_5$. The values of $\beta _2^D$ and $\beta _3^D$ become close in the strong compression region as the flow compressibility increases, which implies that the highly compressible structures are primarily sheet like, and parallel to the plane formed by $\varLambda _2^D$ and $\varLambda _3^D$.

Figure 9. The PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^D$. (a–c) Case $\textrm {I}_3$ and (d–f) case $\textrm {I}_5$.

Figure 10. The PDFs and conditional PDFs of the normalized eigenvalues of ${\mathsf{D}}_{ij}^D$. (a–c) Case $\textrm {II}_3$ and (d–f) case $\textrm {II}_5$.

Figures 11 and 12 display the PDFs and conditional PDFs of the alignments between the vorticity and eigenvectors of the deviatoric strain-rate components (${\mathsf{D}}_{ij}^D$ and ${\mathsf{D}}_{ij}^S$) for cases $\textrm {I}_3$ and $\textrm {I}_5$, respectively. The alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^S$ are almost identical to those for incompressible turbulence (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Buaria et al. Reference Buaria, Bodenschatz and Pumir2020) and weakly compressible turbulence (Pirozzoli & Grasso Reference Pirozzoli and Grasso2004; Lee et al. Reference Lee, Girimaji and Kerimo2009). There is a strong tendency for the vorticity to be perpendicular to the first eigenvector and align with the intermediate eigenvector (figures 11(a,b) and 12(a,b)). The PDF and conditional PDFs of $\textrm {cos}(\omega, \varLambda _3^S)$ approximately equal 0.5, implying that the distribution of angle between the vorticity and the third eigenvector is relatively uniform (figures 11(c), 12(c)). Similar observations are made for the $M_t \approx 0.68$ cases (not shown). Thus, the effects of the vibrational relaxation and dilatation on the alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^S$ are negligible. The above observations agree well with the results of tables 5 and 6, where $\textrm {Corr}(\omega ^2, {\mathsf{D}}_{ij}^S{\mathsf{D}}_{ij}^S)$ is found to be independent of the vibrational relaxation.

Figure 11. The PDFs and conditional PDFs of the alignments between the vorticity and the eigenvectors (a–c) $\varLambda _i^S$ and (d–f) $\varLambda _i^D$. Case $\textrm {I}_3$.

Figure 12. The PDFs and conditional PDFs of alignments between the vorticity and the eigenvectors (a–c) $\varLambda _i^S$ and (d–f) $\varLambda _i^D$. Case $\textrm {I}_5$.

Obviously, alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^D$ are closely related to the vibrational relaxation and local dilatation. For case $\textrm {I}_3$, there is a tendency for the vorticity to be perpendicular to the first eigenvector in the strong compressible region, and to strongly align with the first eigenvector in the strong expansion region (figure 11d). Note that the tendency is weakened as the local dilatation decreases. As shown in figure 11(e), the vorticity slightly aligns with the intermediate eigenvector only in the strong expansion region. Contrary to the first eigenvector, the vorticity tends to align with the third eigenvector in the strong compressible region, and to be strongly perpendicular to the third eigenvector in the strong expansion region (figure 11f). However, for case $\textrm {I}_5$, where the relaxation effect is weak, there is no obvious tendency for alignment between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^D$. The PDF and conditional PDFs of $\cos (\omega, \varLambda _i^D)$ approximately equal 0.5 (figure 12d,e,f).

Figure 13 displays the PDFs and conditional PDFs of the alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^D$ for cases $\textrm {II}_3$ and $\textrm {II}_5$. The conditional PDFs of $\cos (\omega, \varLambda _i^D)$ are similar for cases $\textrm {II}_3$ and $\textrm {II}_5$. The vorticity aligns with the first eigenvector in the expansion region, and with the third eigenvector in the compression region. Meanwhile, the vorticity tends to be perpendicular to the first eigenvector in the compression region, and to the third eigenvector in the expansion region. The alignment between the vorticity and the second eigenvector of ${\mathsf{D}}^D_{ij}$ is not clear for both cases $\textrm {II}_3$ and $\textrm {II}_5$ (figure 13b,e). These observations are similar to that for case $\textrm {I}_3$. This suggests that alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^D$ are closely related to the flow compressibility for both the $M_t \approx 0.22$ and $0.68$ cases. The stronger the flow compressibility, the less significant the alignments between the vorticity and eigenvectors of ${\mathsf{D}}_{ij}^D$.

Figure 13. The PDFs and conditional PDFs of alignments between the vorticity and the eigenvector $\varLambda _i^D$. (a–c) Case $\textrm {II}_3$ and (d–f) case $\textrm {II}_5$.

Figures 14 and 15 display the conditional averages of the enstrophy production along the three eigendirections of ${\mathsf{D}}_{ij}^S$ and ${\mathsf{D}}_{ij}^D$ for the $M_t \approx 0.22$ and $0.68$ cases, respectively. As shown in figure 14(a,b,c), the conditional averages of $\omega ^2 \lambda _1^S \cos ^2(\omega,\varLambda _1^S)/\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are negative, while those of $\omega ^2 \lambda _2^S \cos ^2(\omega,\varLambda _2^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $\omega ^2 \lambda _3^S \cos ^2(\omega,\varLambda _3^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are positive in both compression and expansion regions. The ratio of the conditional averages of $\omega ^2 \lambda _i^S \cos ^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ is close to $-$0.33:0.58:0.66 at $\theta /\theta ^\prime = 0.0$. The magnitude of the conditional average of $\omega ^2 \lambda _1^S \cos ^2(\omega,\varLambda _1^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ for each case is much smaller than those of the other two components. Furthermore, as mentioned above, the PDFs and conditional PDFs of $\beta _i^S$ and $\cos (\omega,\varLambda ^S_i)$ are nearly independent of the local dilatation and vibrational relaxation. Therefore, the variations of $\omega ^2 \lambda _i^S \cos ^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are thus mainly affected by the vorticity. Here, we can observe that the dependencies of magnitudes of $\omega ^2 \lambda _i^S \textrm {cos}^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ on the local dilatation are similar to those of $\omega ^2/\langle \omega ^2 \rangle$ (figure 3a).

Figure 14. Conditional averages of the enstrophy production components along the three eigendirections of the deviatoric strain-rate components (a–c) ${\mathsf{D}}_{ij}^S$ and (d–f) ${\mathsf{D}}_{ij}^D$. Here, $M_t \approx 0.22$.

Figure 15. Conditional averages of enstrophy production components along the three eigendirections of the deviatoric strain-rate components (a–c) ${\mathsf{D}}_{ij}^S$ and (d–f) ${\mathsf{D}}_{ij}^D$. Here, $M_t \approx 0.68$.

Along the eigendirections of ${\mathsf{D}}_{ij}^D$, the conditional averages of $\omega ^2 \lambda _1^D \cos ^2(\omega,\varLambda _1^D)/ \langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $\omega ^2 \lambda _3^D \cos ^2(\omega,\varLambda _3^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are negative and positive, respectively; while those of $\omega ^2 \lambda _2^D \cos ^2(\omega,\varLambda _2^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are positive in the compression region and negative in the expansion region (figure 14d,e,f). Their magnitudes are enlarged with the increase of dilatation, and their growth rates are weakened by the relaxation effect. When the relaxation effect is significant (e.g. case $\textrm {I}_3$), the magnitudes of conditional averages of $\omega ^2 \lambda _1^D \cos ^2(\omega,\varLambda _1^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ in the compression region and $\omega ^2 \lambda _3^D \cos ^2(\omega,\varLambda _3^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ in the expansion region are close to zero and almost remain constant. This is due to the fact that the conditional average of $\cos (\omega,\varLambda _1^D)$ in the compression region and that of $\cos (\omega,\varLambda _3^D)$ in the expansion region tend to be zero (figure 11d,f). The above observations suggest that, different from $\omega ^2 \lambda _i^S \cos ^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, the dependency of $\omega ^2 \lambda _i^D \cos ^2(\omega,\varLambda _i^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ on the local dilatation is affected by the vorticity, eigenvalues and alignments between the vorticity and strain eigenvectors.

For the $M_t \approx 0.68$ cases, as presented in figure 15(a,b,c), the conditional averages of $\omega ^2 \lambda _1^S \cos ^2(\omega,\varLambda _1^S)/\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are negative in both compression and expansion regions, while those of $\omega ^2 \lambda _2^S \cos ^2(\omega,\varLambda _2^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $\omega ^2 \lambda _3^S \cos ^2(\omega,\varLambda _3^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are positive. The ratio of conditional averages of $\omega ^2 \lambda _i^S \cos ^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ is close to $-$0.25:0.57:0.54 at $\theta /\theta ^\prime = 0.0$. Their magnitudes almost remain constant in the range of $\theta /\theta ^\prime < -4.0$, decline in the range of $-4.0 \leq \theta /\theta ^\prime \leq 0.0$ and increase sharply in the expansion region. The influence of vibrational relaxation is mainly manifested in the strong expansion region, especially for the intermediate eigendirection. It can be expected that the conditional averages of $\omega ^2 \lambda _i^S \cos ^2(\omega,\varLambda _i^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ mainly relate to the vorticity. Note that the magnitude of the conditional average of $\omega ^2 \lambda _1^S \cos ^2(\omega,\varLambda _1^S) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ for each case is approximately half that of the other two components. On the other hand, the conditional averages of $\omega ^2 \lambda _1^D \cos ^2(\omega,\varLambda _1^D)/\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ and $\omega ^2 \lambda _3^D \cos ^2(\omega,\varLambda _3^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are respectively negative and positive in both compression and expansion regions; while those of $\omega ^2 \lambda _2^D \cos ^2(\omega,\varLambda _2^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ are positive in the compression region and negative in the expansion region (figure 15d,e,f). The magnitudes of conditional averages of $\omega ^2 \lambda _i^D \cos ^2(\omega,\varLambda _i^D)/\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ along three eigendirections are close to zero at $\theta /\theta ^\prime = 0.0$. Their magnitudes are enlarged with the increase of dilatation, and their growth rates are weakened by the relaxation effect. Meanwhile, the magnitude of the conditional average of $\omega ^2 \lambda _1^D \cos ^2(\omega,\varLambda _1^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ for each case is slightly larger than that of $\omega ^2 \lambda _3^D \cos ^2(\omega,\varLambda _3^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$, while that of $\omega ^2 \lambda _2^D \cos ^2(\omega,\varLambda _2^D) /\langle \omega _i \omega _j {\mathsf{S}}_{ij}^* \rangle$ is much smaller. The above observations are consistent with the results in tables 11 and 12.