Isotropic turbulence of variable-density incompressible flows

In the present study, the effects of density variations on structures developing in an isotropic incompressible turbulent flow are investigated. Statistical analyses are carried out on data sets obtained from direct numerical simulations of forced turbulence. The discretized variable-density incompressible Navier-Stokes equations are time advanced with a Fourier-Fourier spectral solver coupled with a semi-implicit second order in time Runge-Kutta scheme. Turbulence is forced using an extension of the Lundgren method to the variable-density equations including mass diffusion effects. Numerical evidence shows that the introduction of a variable-density field into a turbulent field modifies the coherent structures and the energy spectrum in the inertial range. The analysis of probability density functions of velocity gradients and Lagrangian acceleration suggests an increase in time and space intermittency beyond a threshold density ratio associated with the two-fluid mixture. These modifications are not captured by the classical scaling laws of skewness and flatness factors given in the literature for the constant-density flow case. The energy spectra preserve the Kolmogorov slope while exhibiting an energy level alteration within the smallest scales of the inertial range. This region corresponds to the range of modes where the energy levels of the Rayleigh-Taylor instability criterion are the highest, giving some physical arguments that the aforementioned structural modifications may be attributed to Rayleigh-Taylor-like instabilities.

Figure 1

Energy spectra at final time $t=t_f$ of constant-density HIT run 256-1 used as initial condition for the variable-density HIT simulations. The vertical line delimits the mode associated with the aliasing filter. Simulation parameters for the constant-density HIT DNS are listed in Table 2.

Figure 2

$xz$ cross-section plot of the density field at different times during simulation run 256-4. $\mathrm{Re}=1000, s=4, \mathrm{Sc}=1, N=256$.

Figure 3

Evolution of the volumetric total kinetic energy, isotropy degree [33] (calculation detailed in Appendix pp2), microscale Reynolds number ${\mathrm{Re}}_{\lambda }$, density field standard deviation in space $\sigma \left(\rho \right)$, velocity longitudinal-derivative skewness $S$, and eddy turnover time $T$ during simulation run 256-4. Vertical lines delimit a time interval $\mathrm{\Gamma }$ of two eddy turnover times, where turbulent parameters do not fluctuate much and where mass diffusion effects are still significant. $\mathrm{Re}=1000, s=4, \mathrm{Sc}=1, N=256$.

Figure 4

Density field PDF evolution during DNSs of variable-density HIT. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 5

Density field moments evolution during DNSs of variable-density HIT: (left) skewness defined in Eq. (19); (right) flatness defined in Eq. (20). Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 6

$\frac{\partial \rho }{\partial x}$ PDF evolution during DNSs of variable-density HIT. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 7

$\frac{\partial \rho }{\partial y}$ PDF evolution during DNSs of variable-density HIT. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 8

Time evolution of the turbulent Reynolds number ${\mathrm{Re}}_{\lambda }=(\lambda /L)\mathrm{Re}$ during variable-density simulations. DNS parameters are listed in Table 1. For comparison, dashed line depicts the evolution throughout constant-density simulation 256-1 for $t\in [40,50]$. Vertical lines delimit the study time interval $\mathrm{\Gamma }$, which is considered for the statistical computations.

Figure 9

$\zeta$ PDFs for (a), (b) $\zeta =\frac{\partial u}{\partial x}$, (c) $\zeta =\frac{\partial v}{\partial x}$, (d) $\zeta ={\omega }_x$, (e) $\zeta ={\omega }_y$, (f) $\zeta =A_x$ (Lagrangian acceleration), averaged over the study time interval $\mathrm{\Gamma }$ of variable-density simulations, and compared with the constant-density initial condition (dashed). DNS parameters are listed in Table 1. $\sigma$ is the standard deviation of $\zeta$ in each PDF. $\langle {\mathrm{Re}}_{\lambda }\rangle$ represents the averaged turbulent Reynolds number over time interval $\mathrm{\Gamma }$. In (b) the PDF has not been normalized by $\sigma$.

Figure 10

Skewness ($S$) and flatness ($F$) coefficients from variable-density HIT DNSs compared with constant-density HIT scaling laws (dashed), Eqs. (23) and (24), proposed by Ishihara et al. [13].

Figure 11

$xz$ cross-section plot of the vorticity norm for both (a) and (b) the constant-density initial condition and (c) and (d) the variable-density simulation 256-4 (at time $t=4$). $\mathrm{Re}=1000, s=4, \mathrm{Sc}=1, N=256$.

Figure 12

Spectra averaged over the study time interval $\mathrm{\Gamma }$ of variable-density simulations, and compared with the constant-density initial condition (dashed). Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 13

Spectra averaged over the study time interval $\mathrm{\Gamma }$ of variable-density simulations, and compared with the constant-density initial condition (dashed). Dotted lines correspond to spectra $E_{\rho }$ computed with kinetic energy per unit of volume $\rho \mathit{u}$. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 14

Spectra averaged over the study time interval $\mathrm{\Gamma }$ of variable-density simulations and compared with the constant-density initial condition (dashed). The vertical lines represent the integral and Taylor scales associated with the equations defined in Eqs. (16) and (17). Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 15

$xz$ cross-section plot of the Rayleigh-Taylor instability criterion ($R$) at time $t=4$ of simulation 256-4. $\mathrm{Re}=1000, s=4, \mathrm{Sc}=1, N=256$.

Figure 16

Spectra averaged over the study time interval $\mathrm{\Gamma }$ of variable-density simulations, and compared with the spectral density of the Rayleigh-Taylor instability criterion $R$ (dotted). DNS parameters are listed in Table 1.

Figure 17

Evolution of the relative difference between $\mathcal{E}\left(\mathit{k}\right)$ and $E(k,t)/4\pi k^2$ during simulation run 256-8. The difference between these two terms measures directional anisotropy. Here a difference of a few percent is observed, indicating that the flow is isotropic. A similar behavior is obtained in all simulation runs.

Figure 18

Evolution of the volumetric total kinetic energy, isotropy degree [33] (calculation detailed in Appendix pp2), microscale Reynolds number ${\mathrm{Re}}_{\lambda }$, density field standard deviation in space $\sigma \left(\rho \right)$, velocity longitudinal-derivative skewness $S$, and eddy turnover time $T$ during simulation run 256-2. Vertical lines delimit the statistical analysis time interval $\mathrm{\Gamma }$. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 19

Evolution of the volumetric total kinetic energy, isotropy degree [33] (calculation detailed in Appendix pp2), microscale Reynolds number ${\mathrm{Re}}_{\lambda }$, density field standard deviation in space $\sigma \left(\rho \right)$, velocity longitudinal-derivative skewness $S$, and eddy turnover time $T$ during simulation run 256-6. Vertical lines delimit the statistical analysis time interval $\mathrm{\Gamma }$. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 20

Evolution of the volumetric total kinetic energy, isotropy degree [33] (calculation detailed in Appendix pp2), microscale Reynolds number ${\mathrm{Re}}_{\lambda }$, density field standard deviation in space $\sigma \left(\rho \right)$, velocity longitudinal-derivative skewness $S$, and eddy turnover time $T$ during simulation run 256-8. Vertical lines delimit the statistical analysis time interval $\mathrm{\Gamma }$. Simulation parameters for variable-density HIT DNSs are listed in Table 1.

Figure 21

Evolution of the volumetric total kinetic energy, isotropy degree [33] (calculation detailed in Appendix pp2), microscale Reynolds number ${\mathrm{Re}}_{\lambda }$, density field standard deviation in space $\sigma \left(\rho \right)$, velocity longitudinal-derivative skewness $S$, and eddy turnover time $T$ during simulation run 256-0.5. Vertical lines delimit the statistical analysis time interval $\mathrm{\Gamma }$. Simulation parameters for variable-density HIT DNSs are listed in Table 1.