Turbulent front speed in the Fisher equation: dependence on Damköhler number

The collective effect of finite-sized inhomogeneities on the spatial spread of populations in two dimensions

The dynamics of a population expanding into unoccupied habitat has been primarily studied for situations in which growth and dispersal parameters are uniform in space or vary in one dimension. Here, we study the influence of finite-sized individual inhomogeneities and their collective effect on front speed if randomly placed in a two-dimensional habitat. We use an individual-based model to investigate the front dynamics for a region in which dispersal or growth of individuals is reduced to zero (obstacles) or increased above the background (hotspots), respectively. In a regime where front dynamics is determined by a local front speed only, a principle of least time can be employed to predict front speed and shape. The resulting analytical solutions motivate an event-based algorithm illustrating the effects of several obstacles or hotspots. We finally apply the principle of least time to large heterogeneous environments by solving the Eikonal equation numerically. Obstacles lead to a slow-down that is dominated by the number density and width of obstacles, but not by their precise shape. Hotspots result in a speed-up, which we characterize as function of hotspot strength and density. Our findings emphasize the importance of taking the dimensionality of the environment into account.

Turbulent diffusion of chemically reacting flows: Theory and numerical simulations

The theory of turbulent diffusion of chemically reacting gaseous admixtures developed previously [T. Elperin et al., Phys. Rev. E 90, 053001 (2014)PLEEE81539-375510.1103/PhysRevE.90.053001] is generalized for large yet finite Reynolds numbers and the dependence of turbulent diffusion coefficient on two parameters, the Reynolds number and Damköhler number (which characterizes a ratio of turbulent and reaction time scales), is obtained. Three-dimensional direct numerical simulations (DNSs) of a finite-thickness reaction wave for the first-order chemical reactions propagating in forced, homogeneous, isotropic, and incompressible turbulence are performed to validate the theoretically predicted effect of chemical reactions on turbulent diffusion. It is shown that the obtained DNS results are in good agreement with the developed theory.

Front structure and dynamics in dense colonies of motile bacteria: Role of active turbulence

We study the spreading of a bacterial colony undergoing turbulentlike collective motion. We present two minimalistic models to investigate the interplay between population growth and coherent structures arising from turbulence. Using direct numerical simulation of the proposed models we find that turbulence has two prominent effects on the spatial growth of the colony: (a) the front speed is enhanced, and (b) the front gets crumpled. Both these effects, which we highlight by using statistical tools, are markedly different in our two models. We also show that the crumpled front structure and the passive scalar fronts in random flows are related in certain regimes.

Turbulent diffusion of chemically reacting gaseous admixtures

We study turbulent diffusion of chemically reacting gaseous admixtures in a developed turbulence. In our previous study [Phys. Rev. Lett. 80, 69 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.69] using a path-integral approach for a delta-correlated in a time random velocity field, we demonstrated a strong modification of turbulent transport in fluid flows with chemical reactions or phase transitions. In the present study we use the spectral τ approximation that is valid for large Reynolds and Peclet numbers and show that turbulent diffusion of the reacting species can be strongly depleted by a large factor that is the ratio of turbulent and chemical times (turbulent Damköhler number). We have demonstrated that the derived theoretical dependence of a turbulent diffusion coefficient versus the turbulent Damköhler number is in good agreement with that obtained previously in the numerical modeling of a reactive front propagating in a turbulent flow and described by the Kolmogorov-Petrovskii-Piskunov-Fisher equation. We have found that turbulent cross-effects, e.g., turbulent mutual diffusion of gaseous admixtures and turbulent Dufour effect of the chemically reacting gaseous admixtures, are less sensitive to the values of stoichiometric coefficients. The mechanisms of the turbulent cross-effects differ from the molecular cross-effects known in irreversible thermodynamics. In a fully developed turbulence and at large Peclet numbers the turbulent cross-effects are much larger than the molecular ones. The obtained results are applicable also to heterogeneous phase transitions.

Measurement of large spiral and target waves in chemical reaction-diffusion-advection systems: turbulent diffusion enhances pattern formation

In the absence of advection, reaction-diffusion systems are able to organize into spatiotemporal patterns, in particular spiral and target waves. Whenever advection is present that can be parametrized in terms of effective or turbulent diffusion D(*), these patterns should be attainable on a much greater, boosted length scale. However, so far, experimental evidence of these boosted patterns in a turbulent flow was lacking. Here, we report the first experimental observation of boosted target and spiral patterns in an excitable chemical reaction in a quasi-two-dimensional turbulent flow. The wave patterns observed are ~50 times larger than in the case of molecular diffusion only. We vary the turbulent diffusion coefficient D(*) of the flow and find that the fundamental Fisher-Kolmogorov-Petrovsky-Piskunov equation, v(f) proportional sqrt[D(*)], for the asymptotic speed of a reactive wave remains valid. However, not all measures of the boosted wave scale with D(*) as expected from molecular diffusion, since the wave fronts turn out to be highly filamentous.