arXiv subjects

Charles R. Doering

Publications and source records attributed to Charles R. Doering.

47 records · Page 3Linked to original sources

arXiv2008

Mixing effectiveness depends on the source-sink structure: Simulation results

The mixing effectiveness, i.e., the enhancement of molecular diffusion, of a flow can be quantified in terms of the suppression of concentration variance of a passive scalar sustained by steady sources and sinks. The mixing enhancement defined this way is the ratio of the RMS fluctuations of the scalar mixed by molecular diffusion alone to the (statistically steady-state) RMS fluctuations of the scalar density in the presence of stirring. This measure of the effectiveness of the stirring is naturally related to the enhancement factor of the equivalent eddy diffusivity over molecular diffusion, and depends on the Peclet number. It was recently noted that the maximum possible mixing enhancement at a given Peclet number depends as well on the structure of the sources and sinks. That is, the mixing efficiency, the effective diffusivity, or the eddy diffusion of a flow generally depends on the sources and sinks of whatever is being stirred. Here we present the results of particle-based simulations quantitatively confirming the source-sink dependence of the mixing enhancement as a function of Peclet number for a model flow.

physics.flu-dyn↗

arXiv2006

Multiscale Mixing Efficiencies for Steady Sources

Multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales. We consider scalars maintained by temporally steady but spatially inhomogeneous sources, stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence. The mixing efficiencies are rigorously bounded in terms of the Peclet number and specific quantitative features of the source. Scaling exponents for the bounds at high Peclet number depend on the spectrum of length scales in the source, indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories.

physics.flu-dyn↗

arXiv2006

Energy Dissipation in Fractal-Forced Flow

The rate of energy dissipation in solutions of the body-forced 3-d incompressible Navier-Stokes equations is rigorously estimated with a focus on its dependence on the nature of the driving force. For square integrable body forces the high Reynolds number (low viscosity) upper bound on the dissipation is independent of the viscosity, consistent with the existence of a conventional turbulent energy cascade. On the other hand when the body force is not square integrable, i.e., when the Fourier spectrum of the force decays sufficiently slowly at high wavenumbers, there is significant direct driving at a broad range of spatial scales. Then the upper limit for the dissipation rate may diverge at high Reynolds numbers, consistent with recent experimental and computational studies of "fractal-forced'' turbulence.

physics.flu-dyn↗

arXiv2006

Energy and enstrophy dissipation in steady state 2-d turbulence

Upper bounds on the bulk energy dissipation rate $ε$ and enstrophy dissipation rate $χ$ are derived for the statistical steady state of body forced two dimensional turbulence in a periodic domain. For a broad class of externally imposed body forces it is shown that $ε\le k_{f} U^3 Re^{-1/2}(C_1+C_2 Re^{-1})^{1/2}$ and $χ\le k_{f}^{3}U^3 (C_1+C_2 Re^{-1})$ where $U$ is the root-mean-square velocity, $k_f$ is a wavenumber (inverse length scale) related with the forcing function, and $Re = U /νk_f$. The positive coefficients $C_1$ and $C_2$ are uniform in the the kinematic viscosity $ν$, the amplitude of the driving force, and the system size. We compare these results with previously obtained bounds for body forces involving only a single length scale, or for velocity dependent a constant-energy-flux forces acting at finite wavenumbers. Implications of our results are discussed.

physics.flu-dyn↗

arXiv2006

Links between dissipation, intermittency, and helicity in the GOY model revisited

High-resolution simulations within the GOY shell model are used to study various scaling relations for turbulence. A power-law relation between the second-order intermittency correction and the crossover from the inertial to the dissipation range is confirmed. Evidence is found for the intermediate viscous dissipation range proposed by Frisch and Vergassola. It is emphasized that insufficient dissipation-range resolution systematically drives the energy spectrum towards statistical-mechanical equipartition. In fully resolved simulations the inertial-range scaling exponents depend on both model parameters; in particular, there is no evidence that the conservation of a helicity-like quantity leads to universal exponents.

nlin.CD↗

arXiv2004

A Bound on Mixing Efficiency for the Advection-Diffusion Equation

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional "shape" of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.

physics.flu-dyn↗

arXiv2004

Variational bounds on the energy dissipation rate in body-forced shear flow

A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations.

physics.flu-dyn↗

arXiv2004

Intermittency and regularity issues in 3D Navier-Stokes turbulence

Two related open problems in the theory of 3D Navier-Stokes turbulence are discussed in this paper. The first is the phenomenon of intermittency in the dissipation field. Dissipation-range intermittency was first discovered experimentally by Batchelor and Townsend over fifty years ago. It is characterized by spatio-temporal binary behaviour in which long, quiescent periods in the velocity signal are interrupted by short, active `events' during which there are violent fluctuations away from the average. The second and related problem is whether solutions of the 3D Navier-Stokes equations develop finite time singularities during these events. This paper shows that Leray's weak solutions of the three-dimensional incompressible Navier-Stokes equations can have a binary character in time. The time-axis is split into `good' and `bad' intervals: on the `good' intervals solutions are bounded and regular, whereas singularities are still possible within the `bad' intervals. An estimate for the width of the latter is very small and decreases with increasing Reynolds number. It also decreases relative to the lengths of the good intervals as the Reynolds number increases. Within these `bad' intervals, lower bounds on the local energy dissipation rate and other quantities, such as $\|\bu(\cdot, t)\|_{\infty}$ and $\|\nabla\bu(\cdot, t)\|_{\infty}$, are very large, resulting in strong dynamics at sub-Kolmogorov scales. Intersections of bad intervals for $n\geq 1$ are related to Scheffer's potentially singular set in time. It is also proved that the Navier-Stokes equations are conditionally regular provided, in a given `bad' interval, the energy has a lower bound that is decaying exponentially in time.

math.DS↗

arXiv2004

Low-wavenumber forcing and turbulent energy dissipation

A popular method of forcing the fluid in Direct Numerical Simulations of turbulence is to take the body force proportional to the projection of the velocity of the fluid onto its lowest Fourier modes, while keeping the injected external power constant. In this paper we perform a simple but rigorous analysis to establish bounds on the relationship between the energy dissipation rate and the resulting Reynolds number for this type of forcing. While this analysis cannot give detailed information of the energy spectrum, it does provide some indication of the balance of energy between the lower, directly forced, modes, and those excited by the cascade.

physics.flu-dyn↗

arXiv2004

Extinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation

We consider extinction times for a class of birth-death processes commonly found in applications, where there is a control parameter which determines whether the population quickly becomes extinct, or rather persists for a long time. We give an exact expression for the discrete case and its asymptotic expansion for large values of the population. We have results below the threshold, at the threshold, and above the threshold (where there is a quasi-stationary state and the extinction time is very long.) We show that the Fokker-Planck approximation is valid only quite near the threshold. We compare our analytical results to numerical simulations for the SIS epidemic model, which is in the class that we treat. This is an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.

q-bio.PE↗

arXiv2003

Energy dissipation in body-forced plane shear flow

We study the problem of body-force driven shear flows in a plane channel of width l with free-slip boundaries. A mini-max variational problem for upper bounds on the bulk time averaged energy dissipation rate epsilon is derived from the incompressible Navier-Stokes equations with no secondary assumptions. This produces rigorous limits on the power consumption that are valid for laminar or turbulent solutions. The mini-max problem is solved exactly at high Reynolds numbers Re = U*l/nu, where U is the rms velocity and nu is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor beta=epsilon*l/U^3 that depends only on the ``shape'' of the shearing body force. For a simple half-cosine force profile, for example, the high Reynolds number bound is beta <= pi^2/sqrt{216} = .6715... . We also report extensive direct numerical simulations for this particular force shape up to Re approximately 400; the observed dissipation rates are about a factor of three below the rigorous high-Re bound. Interestingly, the high-Re optimal solution of the variational problem bears some qualitative resemblence to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in J. Fluid Mech. 467, 289-306 (2002).

nlin.CD↗