Compressible turbulent mixing: Effects of compressibility

We studied by numerical simulations the effects of compressibility on passive scalar transport in stationary compressible turbulence. The turbulent Mach number varied from zero to unity. The difference in driven forcing was the magnitude ratio of compressive to solenoidal modes. In the inertial range, the scalar spectrum followed the $k^{-5/3}$ scaling and suffered negligible influence from the compressibility. The growth of the Mach number showed (1) a first reduction and second enhancement in the transfer of scalar flux; (2) an increase in the skewness and flatness of the scalar derivative and a decrease in the mixed skewness and flatness of the velocity-scalar derivatives; (3) a first stronger and second weaker intermittency of scalar relative to that of velocity; and (4) an increase in the intermittency parameter which measures the intermittency of scalar in the dissipative range. Furthermore, the growth of the compressive mode of forcing indicated (1) a decrease in the intermittency parameter and (2) less efficiency in enhancing scalar mixing. The visualization of scalar dissipation showed that, in the solenoidal-forced flow, the field was filled with the small-scale, highly convoluted structures, while in the compressive-forced flow, the field was exhibited as the regions dominated by the large-scale motions of rarefaction and compression.

Figure 1

Compensated spectrum of kinetic energy according to the Kolmogorov variables.

Figure 2

Compensated spectrum of scalar according to the Obukhov-Corrsin variables.

Figure 3

(a) Compensated spectrum of kinetic energy according to the Kolmogorov variables, from cases of R1S, R2S, and R5S. (b) Compensated spectrum of scalar according to the Obukhov-Corrsin variables, from cases of R1S, R2S, and R5S.

Figure 4

One-dimensional compensated spectrum of scalar according to the Obukhov-Corrsin variables.

Figure 5

Obukhov-Corrsin scaling of second-order structure function of scalar.

Figure 6

Yaglom scaling of mixed third-order velocity-scalar structure function.

Figure 7

Two-dimensional isocontour lines of scalar field from (a) R1S and (b) R4S, in $z=\pi /2$.

Figure 8

(a) Skewness of scalar increment as a function of $r/\eta$. (b) Flatness of scalar increment as a function of $r/\eta$.

Figure 9

(a) ESS structure functions of scalar from R1S (dashed lines) and R4S (solid lines), where the symbols of deltas, gradients, circles, squares, and diamonds are for $p=2$, 4, 6, 8, and 10, respectively. (b) ESS mixed structure functions of velocity scalar from R1S (dashed lines) and R4S (solid lines), where the symbols of gradients, circles, and squares are for $p=6$, 9, and 12, respectively.

Figure 10

Scaling exponents of structure function of scalar against the order number $p$.

Figure 11

Scaling exponents of mixed structure function of velocity scalar against the order number $p$.

Figure 12

Two-dimensional contours of logarithm of scalar dissipation rate from (a) R1S and (b) R4S, in $z=\pi /2$, where the logarithmic base is 10.

Figure 13

The one-point PDF of the normalized scalar dissipation rate.

Figure 14

Autocorrelation of scalar dissipation rate, as a function of $r/\eta$.

Figure 15

Mixing time scale of scalar as a function of the turbulent Mach number. R1S: delta, R2S: gradient, R3S: circle, R4S: square, R3Ca: right-pointing triangle, and R3Cb: left-pointing triangle.

Figure 16

Compensated spectrum of scalar according to the Obukhov-Corrsin variables, from cases of R3S, R3Ca, and R3Cb.

Figure 17

Obukhov-Corrsin scaling of second-order structure function of scalar, from the cases of R3S, R3Ca, and R3Cb.

Figure 18

Two-dimensional contours of scalar-dilatation coupling term of the scalar variance equation from (a) R3S and (b) R3Cb, in $z=\pi /2$.

Figure 19

Two-dimensional contours of logarithm of the dissipation term of the scalar variance equation from (a) R3S and (b) R3Cb, in $z=\pi /2$, where the logarithmic base is 10.

Figure 20

Autocorrelation of scalar dissipation rate, as a function of $r/\eta$, from the cases of R3S, R3Ca, and R3Cb.