Fluctuations of a passive scalar in a turbulent mixing layer

The turbulent flow originating downstream of the Kelvin-Helmholtz instability in a mixing layer has great relevance in many applications, ranging from atmospheric physics to combustion in technical devices. The mixing of a substance by the turbulent velocity field is usually involved. In this paper, a detailed statistical analysis of fluctuations of a passive scalar in the fully developed region of a turbulent mixing layer from a direct numerical simulation is presented. Passive scalar spectra show inertial ranges characterized by scaling exponents $-4/3$ and $-3/2$ in the streamwise and spanwise directions, in agreement with a recent theoretical analysis of passive scalar scaling in shear flows [Celani et al., J. Fluid Mech. 523, 99 (2005)]. Scaling exponents of high-order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ${\zeta }_{\infty }=0.4$ at large orders. Saturation of intermittency is confirmed by the self-similarity of the tails of the probability density functions of the scalar increments at different scales $r$ with the scaling factor $r^{-{\zeta }_{\infty }}$ and by the analysis of the cumulative probability of large fluctuations. Conversely, intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient. Probability density functions of the scalar increments in the three directions are compared to assess anisotropy.

Figure 1

Schematic description of the flow configuration. The inlet is on the left ($x=0$) and the outlet on the right ($x=473{\delta }_{\omega ,0}$). The statistical analysis of scalar fluctuations was performed in the region marked by the white boxes: $350{\delta }_{\omega ,0}<x<425{\delta }_{\omega ,0}$ and $y=-9{\delta }_{\omega ,0}<y<-{\delta }_{\omega ,0}$. A two-dimensional cut of the scalar field, ranging from $\Phi ={\Phi }_2$ (dark gray) to $\Phi ={\Phi }_1$ (light gray), is shown in the bottom figure. The top figure shows an enlarged view of the squared gradient of the scalar field (scalar dissipation) in the region analyzed. Light (dark) gray corresponds to large (small) values of the gradient.

Figure 2

Mean velocity $(\langle U\rangle -U_c)/\Delta U$ (open squares) scaled with the convective velocity $U_c=(U_1+U_2)/2$ and velocity difference $\Delta U=U_1-U_2$, velocity fluctuation variance $\langle uu\rangle /\Delta U^2$ (open circles), mean passive scalar $(\langle \Phi \rangle -{\Phi }_c)/\Delta \Phi$ (closed squares) scaled with ${\Phi }_c=({\Phi }_1+{\Phi }_2)/2$ and $\Delta \Phi ={\Phi }_1-{\Phi }_2$, and passive scalar fluctuation variance ${\sigma }^2=\langle \varphi \varphi \rangle /\Delta {\Phi }^2$ (closed circles).

Figure 3

Passive scalar spectra in the streamwise $x$ (solid lines) and spanwise $z$ (dashed lines) directions. The straight dotted lines represent the predictions of equations (6) and (7). The vertical straight line indicates the shear length scale $L_s\approx 40\eta$. The inset shows the streamwise and spanwise spectra multiplied by $k^{4/3}$ and $k^{3/2}$, respectively. The spectra compensated by the classical scaling $k^{-5/3}$ are also shown for comparison.

Figure 4

Second-order structure function (circles) in the (a) streamwise $x$ and (b) spanwise $z$ directions. In the $x$ direction, the structure function calculated using time signals and Taylor's hypothesis is also shown (solid line). The dashed lines show the scaling in the viscous and inertial ranges. The insets show the logarithmic derivative of the structure function with respect to separation. The dashed lines indicate the Kolmogorov-Obukhov-Corrsin dimensional scaling ($2/3$) and the anomalous scaling exponents ($1/3$ and $1/2$) consistent with those of the spectra. The vertical solid lines mark the regions with scaling exponent in agreement with the spectra.

Figure 5

Scaling of passive scalar structure functions for increments in the streamwise direction $x$. The relative logarithmic derivative $d{\log }_{10}{S}_{x,n}\left(r\right)/d{\log }_{10}{S}_{x,2}\left(r\right)$ of structure functions of order 4 (open squares), 6 (closed squares), and 8 (open circles) is shown. The relative logarithmic derivatives are approximately constant over the range $120\eta <r<280\eta$. The inset shows values of the scaling exponents, evaluated as the average of the logarithmic derivatives over the range $120\eta <r<280\eta$. Error bars are estimated from the fluctuations of the logarithmic derivatives with respect to the average over the range $120\eta <r<280\eta$. The dashed line in the inset is the Kolmogorov-Obukhov-Corrsin dimensional scaling [22, 23]. The horizontal line in the main figure indicates the asymptotic value ${\gamma }_{\infty ,2}=1.2$.

Figure 6

Scaling of passive scalar structure functions for increments in the spanwise direction $z$. The relative logarithmic derivative $d{\log }_{10}{S}_{z,n}\left(r\right)/d{\log }_{10}{S}_{z,2}\left(r\right)$ of structure functions of order 4 (open squares), 6 (closed squares), and 8 (open circles) is shown. The horizontal lines identify the plateaus characterizing constant scaling ranges. The inset shows values of the scaling exponents (closed symbols), evaluated as the average of the logarithmic derivatives over the range $100\eta <r<200\eta$, and scaling exponents from the DNS data by Gotoh et al. [57] for homogeneous isotropic turbulence advecting a passive scalar injected using a mean gradient (open symbols). Error bars are estimated from the fluctuations of the logarithmic derivatives with respect to the averages over the range $100\eta <r<200\eta$. The dashed line in the inset is the Kolmogorov-Obukhov-Corrsin dimensional scaling [22, 23].

Figure 7

(a) Probability density functions of the scalar increments $\delta \varphi$ for values of the separation $r_x$ between $10\eta$ and $300\eta$, rescaled with the factor $r_x^{-{\zeta }_{\infty }}$. The arrow indicates increasing $r_x$. (b) Contribution of the PDFs to the sixth-order moment (the sixth-order structure function) for different separations. In the range of scale $120\eta <r<280\eta$, the relative scaling exponents of structure functions saturate to a constant value (Fig. 5). The value ${\zeta }_{\infty }$ is 0.4.

Figure 8

Cumulative probability of having $|{\delta }_r\varphi |>\kappa \sigma$, with $\kappa =2$, for increments in the streamwise $x$ (squares) and spanwise $z$ (circles) directions. The result for the spanwise direction is shifted in the vertical direction by one decade to improve readability. The inset shows the local slopes, calculated as logarithmic derivatives.

Figure 9

Results obtained from the application of a box covering algorithm to the set of points characterized by fluctuations larger than $2\sigma$. The number of boxes needed to cover the set $N\left(r\right)$ is plotted as a function of the size of the box $r$ (circles). The scaling of $N\left(r\right)$ versus $r$ is the fractal dimension of the set and it is close to the value $D_F=2.6$, in agreement with the value of the asymptotic exponent ${\zeta }_{\infty }=0.4$. The bottom inset shows the absolute value of the local slope (circles), calculated as a logarithmic derivative, corresponding to the fractal dimension of the set. The solid line marks the value $2.6$. The top inset shows an example of an instantaneous, two-dimensional slice of the set of points where fluctuations are larger than $2\sigma$ in the region of the mixing layer considered for the analysis.

Figure 10

Relative logarithmic derivative of the total (open squares), plus (open diamonds), and minus (closed triangles) structure functions for streamwise increments and of the total structure function for spanwise increments (closed circles). The results are shown for structure functions of order (a) 4 and (b) 6.

Figure 11

Probability density functions of scalar increments for separation in the three directions: (a) streamwise $r_x$, (b) crosswise $r_y$, and (c) spanwise $r_z$. The statistics are shown for different separation distances: $2\eta$ (open squares), $5\eta$ (closed squares), $10\eta$ (open circles), $20\eta$ (closed circles), $50\eta$ (open triangles), $100\eta$ (closed triangles), and $150\eta$ (open diamonds) (not shown for $y$ separation). All increments are normalized with the rms at the specific separation so that all curves have zero mean and unit variance. The solid lines represent a Gaussian distribution with zero mean and unit variance.

Figure 12

Skewness (open symbols) and flatness (closed symbols) of the scalar differences for different separations in the three directions: $x$ (squares), $y$ (triangles), and $z$ (circles).