Fast chemical reaction in two-dimensional Navier-Stokes flow: Initial regime

This paper studies an infinitely fast bimolecular chemical reaction in a two-dimensional biperiodic Navier-Stokes flow. The reactants in stoichiometric quantities are initially segregated by infinite gradients. The focus is placed on the initial stage of the reaction characterized by a well-defined one-dimensional material contact line between the reactants. Particular attention is given to the effect of the diffusion $\kappa$ of the reactants. This study is an idealized framework for isentropic mixing in the lower stratosphere and is motivated by the need to better understand the effect of resolution on stratospheric chemistry in climate-chemistry models. Adopting a Lagrangian straining theory approach, we relate theoretically the ensemble mean of the length of the contact line, of the gradients along it, and of the modulus of the time derivative of the space-average reactant concentrations (here called the chemical speed) to the joint probability density function of the finite-time Lyapunov exponent $\lambda$ with two times $\tau$ and $\tilde{\tau }$. The time $1/\lambda$ measures the stretching time scale of a Lagrangian parcel on a chaotic orbit up to a finite time $t$, while $\tau$ measures it in the recent past before $t$, and $\tilde{\tau }$ in the early part of the trajectory. We show that the chemical speed scales like ${\kappa }^{1/2}$ and that its time evolution is determined by rare large events in the finite-time Lyapunov exponent distribution. The case of smooth initial gradients is also discussed. The theoretical results are tested with an ensemble of direct numerical simulations (DNSs) using a pseudospectral model.

Figure 1

Reactant fields in a biperiodic domain ${[-\pi ,\pi ]}^2$. Red (or light gray), positive values, and blue (or dark gray), negative values, show the two reactants A and B. From left to right $\mathrm{Pr}=1,16,128$ and from top to bottom $t/T=1,3,8$. The Prandtl number Pr is defined as the ratio of the viscosity of the fluid to the tracer diffusion. Since the viscosity is fixed, an increasing Pr means a decreasing diffusion. $T$ is the integral time scale of the flow.

Figure 2

Density $P_{\lambda }$ of the finite-time Lyapunov exponents shown at different times between $t=0$ and $t=25T$. As time increases, the density shifts toward smaller values.

Figure 3

Function $G_e(\lambda ,t)$ plotted at different times ($0<t/T<25$). $G_e$ is defined such that $P_{\lambda }$—plotted in Fig. 2—can be written $\propto$$-t{G}_e(\lambda ,t)$ with $\underset{\lambda }{\min }{G}_e=0$. As time increases, the function $G_e$ shifts toward smaller values. We note the asymmetry of $G_e$ and the faster convergence for FTLE larger than their ensemble mean. The time asymptotic form of $G_e$ is the Cramer function $G$ corresponding to the longtime FTLE pdf $P_{\lambda }$. An estimate of $G$ is given by the blue circles, using a method detailed in the text.

Figure 4

Maps of FTLE calculated at different times and displayed at starting locations of trajectories in the biperiodic domain ${[-\pi ,\pi ]}^2$. In the top row are shown the vorticity (left) and the strain (right) at $t=0$. In the following panels, ordered from left to right and top to bottom are shown the FTLE maps for $t/T=1,2,4,8,16,24$.

Figure 5

Ensemble average of length of contact line (infinite initial gradient case). The symbols correspond to ensemble averages of DNS for different Prandtl numbers $\text{Pr}=1,2,4,8,16,32,64$, and 128. The black solid line corresponds to $\langle L\rangle$, as estimated from the FTLE pdf using (15), and the bold blue dashed line corresponds to $L_E$ as estimated using the FTLE pdf with the simplified expression (17). The light red dot-dashed line corresponds to an exponential increase at a rate ${\lambda }_1=\underset{\lambda }{\max }[\lambda -G\left(\lambda \right)]\approx 0.027$, which corresponds to the asymptotic behavior in the inviscid limit (19). The latter has been shifted vertically for clarity. Note the log scale for the $y$ axis.

Figure 6

Probability density function of $1/\tau$ plotted at different times. The equivalent time $\tau$ is defined in ($30$). As the time increases, the density shifts toward smaller values.

Figure 7

Correlation between $\lambda$ and $1/\tau$ as a function of time (top left) and joint pdf of $(\lambda ,1/\tau )$, as estimated from the numerical simulations and plotted at different times $t/T=0.25,2,4,7,12,20\text{,}\text{and}25$.

Figure 8

Ensemble average of gradients advected with contact line, multiplied by $\sqrt{\kappa \pi }/A_0$, in the sharp-gradient case. The symbols correspond to ensemble averages over the 34 DNS members for different Prandtl numbers $\text{Pr}=2,4,16,32,64,128$. The lines correspond to the calculation from the Lagrangian straining properties (LSPs): in solid from ($37$) and in dashed from ($38$), considering a perfectly equilibrated contact line with the flow. The dot-dashed line is $1/\sqrt{\tau }$ and corresponds to ($38$) with $\lambda$ and $\tau$ statistically independent. Note the log scale for the time axis.

Figure 9

Ensemble average of the chemical speed in the sharp-gradient case divided by the diffusion $\sqrt{\kappa }$. The symbols correspond to numerical results from the 34-member ensemble, for different Prandtl numbers $\text{Pr}=1,2,4,8,64,128$. The solid line (calculation from the LSP) corresponds to ($39$). The exponential increase at a rate ${\lambda }_1=0.027$ (dashed line) corresponds to the expected asymptotic regime of ($39$).

Figure 10

Ensemble average of the chemical speed, in the smooth-gradient case, divided by $\kappa$. The symbols correspond to numerical results from the 34-member ensemble, for different Prandtl numbers $\text{Pr}=2,4,8,128$. The solid line (calculation from the FTLE) correspond to ($42$). The exponential increase at a rate 0.09 corresponds to the expected asymptotic regime of ($42$), as expressed in ($43$) and has been shifted vertically for clarity. Note the log scale for the $y$ axis.