arXiv subjects

Charles R. Doering

Publications and source records attributed to Charles R. Doering.

At least 37 records · Page 2Linked to original sources

arXiv2017

Maximum palinstrophy amplification in the two-dimensional Navier-Stokes equations

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier-Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $\textrm{Re}$ and palinstrophy $\mathcal{P}$ which maximize $d\mathcal{P}/dt$ is constructed by numerically solving suitable optimization problems for a wide range of $\textrm{Re}$ and $\mathcal{P}$, providing numerical evidence for the sharpness of the analytic estimate $d\mathcal{P}/dt \leq \left(a + b\sqrt{\ln\textrm{Re}+c} \, \right) \mathcal{P}^{3/2}$ with respect to both $\textrm{Re}$ and $\mathcal{P}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e., $\sup_{t > 0} \mathcal{P} (t)$, are observed to scale with the magnitude of the initial palinstrophy $\mathcal{P}(0)$ in accord with the corresponding $\textit{a priori}$ estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier-Stokes equation are discussed.

physics.flu-dyn↗

arXiv2016

Bounds for convection between rough boundaries

We consider Rayleigh-Bénard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than ${\cal O}(Ra^{1/2})$ as $Ra \rightarrow \infty$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the ${\cal O}(Ra^{1/2})$ leading term is $0.242 + 2.925\Vert\nabla h\Vert^2$, where $\Vert\nabla h\Vert^2$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.

physics.flu-dyn↗

arXiv2015

Gene expression dynamics with stochastic bursts: exact results for a coarse-grained model

We present a theoretical framework to analyze the dynamics of gene expression with stochastic bursts. Beginning with an individual-based model which fully accounts for the messenger RNA (mRNA) and protein populations, we propose a novel expansion of the master equation for the joint process. The resulting coarse-grained model reduces the dimensionality of the system, describing only the protein population while fully accounting for the effects of discrete and fluctuating mRNA population. Closed form expressions for the stationary distribution of the protein population and mean first-passage times of the coarse-grained model are derived and large-scale Monte Carlo simulations show that the analysis accurately describes the individual-based process accounting for mRNA population, in contrast to the failure of commonly proposed diffusion-type models.

q-bio.MN↗

arXiv2015

Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field

We investigate the behavior of flows, including turbulent flows, driven by a horizontal body-force and subject to a vertical magnetic field, with the following question in mind: for very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2D, with no dependence along the vertical? We first focus on the quasi-static approximation, i.e. the asymptotic limit of vanishing magnetic Reynolds number Rm << 1: we prove that the flow becomes exactly 2D asymptotically in time, regardless of the initial condition and provided the interaction parameter N is larger than a threshold value. We call this property "absolute two-dimensionalization": the attractor of the system is necessarily a (possibly turbulent) 2D flow. We then consider the full-magnetohydrodynamic equations and we prove that, for low enough Rm and large enough N, the flow becomes exactly two-dimensional in the long-time limit provided the initial vertically-dependent perturbations are infinitesimal. We call this phenomenon "linear two-dimensionalization": the (possibly turbulent) 2D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3D attractors may also exist and be attained for strong enough initial 3D perturbations. These results shed some light on the existence of a dissipation anomaly for magnetohydrodynamic flows subject to a strong external magnetic field.

physics.flu-dyn↗

arXiv2014

Wall to Wall Optimal Transport

The calculus of variations is employed to find steady divergence-free velocity fields that maximize transport of a tracer between two parallel walls held at fixed concentration for one of two constraints on flow strength: a fixed value of the kinetic energy or a fixed value of the enstrophy. The optimizing flows consist of an array of (convection) cells of a particular aspect ratio Gamma. We solve the nonlinear Euler-Lagrange equations analytically for weak flows and numerically (and via matched asymptotic analysis in the fixed energy case) for strong flows. We report the results in terms of the Nusselt number Nu, a dimensionless measure of the tracer transport, as a function of the Peclet number Pe, a dimensionless measure of the energy or enstrophy of the flow. For both constraints the maximum transport Nu_{MAX}(Pe) is realized in cells of decreasing aspect ratio Gamma_{opt}(Pe) as Pe increases. For the fixed energy problem, Nu_{MAX} \sim Pe and Gamma_{opt} \sim Pe^{-1/2}, while for the fixed enstrophy scenario, Nu_{MAX} \sim Pe^{10/17} and Gamma_{opt} \sim Pe^{-0.36}. We also interpret our results in the context of certain buoyancy-driven Rayleigh-Benard convection problems that satisfy one of the two intensity constraints, enabling us to investigate how the transport scalings compare with upper bounds on Nu expressed as a function of the Rayleigh number \Ra. For steady convection in porous media, corresponding to the fixed energy problem, we find Nu_{MAX} \sim \Ra and Gamma_{opt} \sim Ra^{-1/2}$, while for steady convection in a pure fluid layer between free-slip isothermal walls, corresponding to fixed enstrophy transport, Nu_{MAX} \sim Ra^{5/12} and Gamma_{opt} \sim Ra^{-1/4}.

physics.flu-dyn↗

arXiv2013

Bounds on Surface Stress Driven Shear Flow

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress driven flow in two and three dimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number $Gr$, the critical $Gr$ for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient $C_f = τ/\bar{u}^2$, where $τ$ is the applied shear stress and $\bar{u}$ is the mean velocity of the fluid at the surface, for flows at higher $Gr$ including developed turbulence: $C_f le 1/32$ in two dimensions and $C_f \le 1/8$ in three dimensions. This analysis rigorously justifies previously computed numerical estimates.

physics.flu-dyn↗

arXiv2013

The small-world effect is a modern phenomenon

The "small-world effect" is the observation that one can find a short chain of acquaintances, often of no more than a handful of individuals, connecting almost any two people on the planet. It is often expressed in the language of networks, where it is equivalent to the statement that most pairs of individuals are connected by a short path through the acquaintance network. Although the small-world effect is well-established empirically for contemporary social networks, we argue here that it is a relatively recent phenomenon, arising only in the last few hundred years: for most of mankind's tenure on Earth the social world was large, with most pairs of individuals connected by relatively long chains of acquaintances, if at all. Our conclusions are based on observations about the spread of diseases, which travel over contact networks between individuals and whose dynamics can give us clues to the structure of those networks even when direct network measurements are not available. As an example we consider the spread of the Black Death in 14th-century Europe, which is known to have traveled across the continent in well-defined waves of infection over the course of several years. Using established epidemiological models, we show that such wave-like behavior can occur only if contacts between individuals living far apart are exponentially rare. We further show that if long-distance contacts are exponentially rare, then the shortest chain of contacts between distant individuals is on average a long one. The observation of the wave-like spread of a disease like the Black Death thus implies a network without the small-world effect.

physics.soc-ph↗

arXiv2011

Stirring up trouble: Multi-scale mixing measures for steady scalar sources

The mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring, relative to the variance in the absence of stirring. These variances can be weighted at various spatial scales, leading to a family of multi-scale mixing measures and efficiencies. We derive a priori estimates on these efficiencies from the advection--diffusion partial differential equation, focusing on a broad class of statistically homogeneous and isotropic incompressible flows. The analysis produces bounds on the mixing efficiencies in terms of the Peclet number, a measure the strength of the stirring relative to molecular diffusion. We show by example that the estimates are sharp for particular source, sink and flow combinations. In general the high-Peclet number behavior of the bounds (scaling exponents as well as prefactors) depends on the structure and smoothness properties of, and length scales in, the scalar source and sink distribution. The fundamental model of the stirring of a monochromatic source/sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis. The large-scale mixing efficiency follows the upper bound scaling (within a logarithm) at high Peclet number but the intermediate and small-scale efficiencies are qualitatively less than optimal. The Peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi-static model.

physics.flu-dyn↗

arXiv2011

Internal heating driven convection at infinite Prandtl number

We derive an improved rigorous bound on the space and time averaged temperature $ $ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries thermally driven by uniform internal heating. A novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non-singular background profile, yielding the estimate $ \geq 0.419[R\log(R)]^{-1/4}$ where $R$ is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved in the appendix.

physics.flu-dyn↗

arXiv2011

"Ultimate state" of two-dimensional Rayleigh-Benard convection between free-slip fixed temperature boundaries

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $Nu \leq 0.2295 Ra^{5/12}$ uniformly in the Prandtl number Pr. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection.

physics.flu-dyn↗

arXiv2010

Models and measures of mixing and effective diffusion

Mixing a passive scalar field by stirring can be measured in a variety of ways including tracer particle dispersion, via the flux-gradient relationship, or by suppression of scalar concentration variations in the presence of inhomogeneous sources and sinks. The mixing efficiency or efficacy of a particular flow is often expressed in terms of enhanced diffusivity and quantified as an effective diffusion coefficient. In this work we compare and contrast several notions of effective diffusivity. We thoroughly examine the fundamental case of a steady sinusoidal shear flow mixing a scalar sustained by a steady sinusoidal source-sink distribution to explore apparent quantitative inconsistencies among the measures. Ultimately the conflicts are attributed to the noncommutative asymptotic limits of large P$\acute{\text{e}}$clet number and large length-scale separation. We then propose another approach, a generalization of Batchelor's 1949 theory of diffusion in homogeneous turbulence, that helps unify the particle dispersion and concentration variance suppression measures.

physics.flu-dyn↗

arXiv2010

Optimal stirring strategies for passive scalar mixing

We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t --> \infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure---when such a flow exists. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the $H^{-1}$ norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.

physics.flu-dyn↗

arXiv2010

Symmetric factorization of the conformation tensor in viscoelastic fluid models

The positive definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.

math.NA↗

arXiv2010

Destabilizing Taylor-Couette flow with suction

We consider the effect of radial fluid injection and suction on Taylor-Couette flow. Injection at the outer cylinder and suction at the inner cylinder generally results in a linearly unstable steady spiralling flow, even for cylindrical shears that are linearly stable in the absence of a radial flux. We study nonlinear aspects of the unstable motions with the energy stability method. Our results, though specialized, may have implications for drag reduction by suction, accretion in astrophysical disks, and perhaps even in the flow in the earth's polar vortex.

physics.flu-dyn↗

arXiv2010

The Mixing efficiency of open flows

Mixing in open incompressible flows is studied in a model problem with inhomogeneous passive scalar injection on an inlet boundary. As a measure of the efficiency of stirring, the bulk scalar concentration variance is bounded and the bound is shown to be sharp at low Peclet number. Although no specific flow saturating the bound at high Peclet number is produced here, the estimate is conjectured to be approached for flows possessing sufficiently sustained chaotic regions.

physics.flu-dyn↗

arXiv2010

Demographic Fluctuations versus Spatial Variation in the Competition between Fast and Slow Dispersers

Dispersal is an important strategy that allows organisms to locate and exploit favorable habitats. The question arises: given competition in a spatially heterogeneous landscape, what is the optimal rate of dispersal? Continuous population models predict that a species with a lower dispersal rate always drives a competing species to extinction in the presence of spatial variation of resources. However, the introduction of intrinsic demographic stochasticity can reverse this conclusion. We present a simple model in which competition between the exploitation of resources and stochastic fluctuations leads to victory by either the faster or slower of two species depending on the environmental parameters. A simplified limiting case of the model, analyzed by closing the moment and correlation hierarchy, quantitatively predicts which species will win in the complete model under given parameters of spatial variation and average carrying capacity.

q-bio.PE↗

arXiv2009

Extreme vorticity growth in Navier-Stokes turbulence

According to statistical turbulence theory, the ensemble averaged squared vorticity rho_E is expected to grow not faster than drho_E/dt ~ rho_E^{3/2}. Solving a variational problem for maximal bulk enstrophy (E) growth, velocity fields were found for which the growth rate is as large as dE/dt ~ E^3. Using numerical simulations with well resolved small scales and a quasi-Lagrangian advection to track fluid subvolumes with rapidly growing vorticity, we study spatially resolved statistics of vorticity growth. We find that the volume ensemble averaged growth bound is satisfied locally to a remarkable degree of accuracy. Elements with dE/dt ~ E^3 can also be identified, but their growth tends to be replaced by the ensemble-averaged law when the intensities become too large.

physics.flu-dyn↗

arXiv2008

A Comparison of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux

We report the results of high resolution direct numerical simulations of two-dimensional Rayleigh-Bénard convection for Rayleigh numbers up to $\Ra=10^{10}$ in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we considered the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law $\Nu \approx 0.138 \times \Ra^{.285}$ with a scaling exponent indistinguishable from $2/7 = .2857...$ above $\Ra = 10^{7}$. The overall flow dynamics for both scenarios, in particular the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers. The findings are compared and contrasted with results of recent three-dimensional simulations.

physics.flu-dyn↗