A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

Zu, Penghe; Chen, Long; Xin, Jack

doi:10.1016/j.physd.2015.07.001

Cited by 18 publications

(16 citation statements)

References 25 publications

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“…This is consistent with numerical computations of S * 11 using alternate methods [16]. This O(1) behavior of S * 11 has been attributed to the Lagrangian chaotic behavior of the flow [16,104].…”

Section: Effective Transport By Advective-diffusionsupporting

confidence: 87%

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Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows

Murphy

Cherkaev

Xin

et al. 2017

Annals of Mathematical Sciences and Applications

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The enhancement in diffusive transport of passive tracer particles by incompressible, turbulent flow fields is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusive process with an effective diffusivity tensor D *. Two different formulations of integral representations for D * were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, with possibly chaotic dynamics, providing rigorous integral representations for D * involving spectral measures of unbounded self-adjoint operators. We prove that the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D * , which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.

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Section: Effective Transport By Advective-diffusionsupporting

confidence: 87%

“…The steady part (cos y, cos x) of the flow is subject to a time-periodic perturbation that causes transition to Lagrangian chaos [16,104]. In the advection dominated regime, we shall compare our computations of the effective diffusivity for the steady θ = 0 and dynamic θ = 1 settings.…”

mentioning

confidence: 99%

Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows

Murphy

Cherkaev

Xin

et al. 2017

Annals of Mathematical Sciences and Applications

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“…As θ increases, the flow trajectories are more and more mixing and chaotic [17]. The Fourier modes for the flow are:…”

Section: Two-dimensional Time-dependent Flowmentioning

confidence: 99%

“…The first term of (1.2) is a steady cellular flow with a π/4 rotation, and the second term is a time periodic perturbation that introduces an increasing amount of disorder in the flow trajectories as θ becomes larger. At θ = 1, it is fully mixing, and empirically sub-diffusive [17]. The flow (1.2) has served as a model of chaotic advection for Rayleigh-Bénard experiment [3].…”

Section: Introductionmentioning

confidence: 99%

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

Lyu

Xin

2017

Numer. Math. Theory Methods Appl.

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We study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bénard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these snapshots as column vectors. The singular values decay rapidly and allow us to extract a small percentage of left singular vectors corresponding to the top singular values as adaptive basis vectors. The trained orthogonal adaptive basis makes possible low cost computation of the effective diffusivities at smaller molecular diffusivities (testing). The testing errors decrease as the training occurs at smaller molecular diffusivities. We make use of the Poincaré map of the advection-diffusion equation to bypass long time simulation and gain accuracy in computing effective diffusivity and learning adaptive basis. We observe a non-monotone relationship between residual diffusivity and the amount of chaos in the advection, though the overall trend is that sufficient chaos leads to higher residual diffusivity.

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“…with θ ω (t) = tan −1 (cos(ωt)). Also the Rayleigh-Bénard advection is known for chaotic streamlines and diffusion-like transport in diagonal direction [3,26]. See figure 1.…”

Section: Introductionmentioning

confidence: 99%

Synchronized Front Propagation and Delayed Flame Quenching in Strain G-equation and Time-Periodic Cellular Flows

Liu¹,

Xin²

2021

Preprint

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G-equations are level-set type Hamilton-Jacobi partial differential equations modeling propagation of flame front along a flow velocity and a laminar velocity. In consideration of flame stretching, strain rate may be added into the laminar speed. We perform finite difference computation of G-equations with the discretized strain term being monotone with respect to one-sided spatial derivatives. Let the flow velocity be the time-periodic cellular flow (modeling Rayleigh-Bénard advection), we compute the turbulent flame speeds as the asymptotic propagation speeds from a planar initial flame front. In strain G-equation model, front propagation is enhanced by the cellular flow, and flame quenching occurs if the flow intensity is large enough. In contrast to the results in steady cellular flow, front propagation in time periodic cellular flow may be locked into certain spatial-temporal periodicity pattern, and turbulent flame speed becomes a piecewise constant function of flow intensity. Also the disturbed flame front does not cease propagating until much larger flow intensity.

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A computational study of residual KPP front speeds in time-periodic cellular flows in the small diffusion limit

Cited by 18 publications

References 25 publications

Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows

Spectral analysis and computation of effective diffusivities in space-time periodic incompressible flows

Computing Residual Diffusivity by Adaptive Basis Learning via Spectral Method

Synchronized Front Propagation and Delayed Flame Quenching in Strain G-equation and Time-Periodic Cellular Flows

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