Comparison of Lagrangian and Eulerian frames of passive scalar turbulent mixing

The mixing of a passive scalar in a three-dimensional, statistically stationary turbulent Navier-Stokes flow at a constant and moderate Taylor microscale Reynolds number $R_{\lambda }=42$ is studied by means of direct numerical simulations for Schmidt numbers between 1 and 64. The freely decaying passive scalar is represented in two different ways: (1) in the Lagrangian frame of reference as a cloud of up to 4.8 billion individually advected massless tracer particles subject to a stochastic Wiener process along the tracer tracks that describes scalar diffusion or (2) in the standard Eulerian frame of reference as an advection-diffusion equation of the continuum concentration field. In both cases, the scalar is initially seeded in a small cubic subvolume. The mean mixing time $\langle t_s\rangle$ is determined by the mean compressive strain rate $\langle {\lambda }_3\rangle <0$ which is obtained from the probability density functions of the local finite-time Lyapunov exponents in the Lagrangian frame, ${\lambda }_i\left(t\right)$ with $i=1,2$ and 3. The direct comparison of freely decaying Lagrangian and Eulerian passive scalars gives a good agreement of the scalar variance for times $t\lesssim 10\langle t_s\rangle$ and for the probability density functions $P(\mathrm{\Theta },t)$ taken with respect to the whole simulation domain. We also show how the multilayer aggregations of scalar filaments and sheets in the Lagrangian frame are increasingly influenced by the noise due to discreteness with progressing dilution of the initially high tracer particle concentration. This limits the Lagrangian approach in its present form and for the obtainable Schmidt numbers to studies of shorter time periods. A simple one-dimensional advection-diffusion model of a solitary strip is finally applied to the problem at hand to derive the probability density function of the scalar concentration, $P(\mathrm{\Theta },t)$, from the one of the compressive local finite-time Lyapunov exponent, $p({\lambda }_3,t)$. Model prediction with and without self-convolution and numerical data of the scalar distributions agree qualitatively, however with quantitative differences particularly for small scalar concentrations. The present Lagrangian approach to passive scalar mixing in turbulence opens the application of more flexible passive scalar injection and boundary conditions and allows to relax the resolution constraints for high-Schmidt number mixing studies.

Figure 1

Snapshot of the advecting turbulent velocity field. Velocity field vectors are projected into the $x-y$ plane at $z=L_z/2$. The arrows are colored by the velocity magnitude which is given in units of the root-mean-square velocity $u_{\mathrm{rms}}$ taken as a combined volume-time average with respect to all three components.

Figure 2

Mean profile of the streamwise velocity field component $u_1$ with respect to $x_2$. The profile is obtained by a combined average over coordinates $x_1$, $x_3$ and time $t$. This flow property is unchanged for all cases discussed below. Amplitude is given in units of the root-mean-square velocity $u_{\mathrm{rms}}$.

Figure 3

Finite-time Lyapunov exponents (FTLE) analysis. (a)–(c) Contour plot of the calculated first FTLE, ${\lambda }_1$, for times at $t=0.8$, $t=2$, and $t=5$ s, the dark color represents zones of high stretching rates or where particle trajectories separate most strongly. (d)–(f): The corresponding probability distributions of all three Lyapunov exponents. The first exponent ${\lambda }_1$ is strictly positive and the third exponent ${\lambda }_3$ is strictly negative. With increasing time the distributions converge slowly to the limit of Gaussian distributions with decreasing width. The dashed red line in panel (f) is the corresponding Gaussian distribution. The legend is the same for panels (d)–(f).

Figure 4

Mixing of a passive scalar in a periodic slab domain. The passive scalar is simulated in the Lagrangian frame of reference as an ensemble of 4.8 billion individual tracers with additional noise (Langevin equation). Initially all tracers start in a small subdomain. We display the configuration for times $t=0\langle t_s\rangle$, $0.55\langle t_s\rangle$, $1.10\langle t_s\rangle$, and $2.20\langle t_s\rangle$. The Schmidt number is $\mathrm{Sc}=64$. The additional shear forcing causes a mean advection in $x_1$ direction.

Figure 5

Variance of scalar concentration fluctuations, $\theta$, versus time. Time is given in units of the mean mixing time $\langle t_s\rangle$. The plot shows the direct comparison of Eulerian and Lagrangian runs for freely decaying scalar turbulence at $\mathrm{Sc}=1$ (left), $\mathrm{Sc}=32$ (middle), and $\mathrm{Sc}=64$ (right). For $t/\langle t_s\rangle \gtrsim 5$ an overlap of the scalar filaments occurs where they re-enter the simulation domain in the downstream direction $x_1$, due to the applied periodic boundary conditions. This overlapping or aggregation causes the change in the decay rate. The four highlighted data points in the middle panel indicate time instants for an analysis which will be discussed in Sec. 4.

Figure 6

Probability density functions (PDFs) of the passive scalar concentration taken at different times of the evolution for different Schmidt numbers. The left column shows the Lagrangian cases L1, L2, L3, the right column the Eulerian cases E1, E2, E3. Top row: $\mathrm{Sc}=1$. Middle row: $\mathrm{Sc}=32$. Bottom row: $\mathrm{Sc}=64$. At the beginning, the scalar PDF is a bimodal distribution with entries at $\mathrm{\Theta }/{\mathrm{\Theta }}_0=0$ and 1 only.

Figure 7

Isocontour plots of the concentration field for the cases L3 and E3 of a decaying passive scalar with Schmidt number $\mathrm{Sc}=64$ at $x=4d$ with $d/L=0.25$. The field is represented by 4.8 billion individual tracer particles (left column) or by an Eulerian scalar field advected by the fluid flow (right column) for times $t=2.75\langle t_s\rangle$ (top row) and $t=5.5\langle t_s\rangle$ (bottom row).

Figure 8

Maximum scalar concentration versus time. Comparison of the approximation (gray curves) and the error function (black curves) for the cases $\text{Sc}=1$ (dashed lines) and $\text{Sc}=32$ (solid lines).

Figure 9

Scalar distribution (red lines) shown together with the distribution based on the prediction from the smallest Lyapunov exponent (solid gray lines) as defined in Eq. (43). The additional gray lines represent (multiple) self-convolution operations of the passive scalar PDFs. Times are $t=1.79$ $\langle t_s\rangle$ in (a), 2.38 $\langle t_s\rangle$ in (b), 2.98 $\langle t_s\rangle$ in (c), and 3.57 $\langle t_s\rangle$ in (d). The insets in the figure replot the corresponding data on logarithmic axes.

Figure 10

Time dependence of the fit coefficient $\alpha$ for the scalar distribution exponential tails which have the shape $exp(-\alpha \mathrm{\Theta })$ as shown in Fig. 6.

Figure 11

Left: Test case for a stochastic Wiener process with different amplitudes $\sqrt{2D}$ which correspond to different Schmidt number. Right: Verification of the Einstein relation for the mean square displacement with increasing number of realizations (or particles) for $\text{Sc}=100$.