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 Péclet number, a measure of 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-Péclet-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 Péclet number but the intermediate and small-scale efficiencies are qualitatively less than optimal. The Péclet 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.     

Scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence

Statistics of a passive scalar flux in a uniform mean scalar gradient convected by homogeneous isotropic steady turbulence are numerically studied by using very high resolution direct numerical simulation. It is found that the Nusselt number increases in proportion to the Péclet number and that the one point probability density function of the scalar flux is negatively skewed and exponential, and is insensitive to the variation of the Péclet number. The scalar field is studied by visualization, and the ramp-cliff structure and the mesa-canyon structure are observed along the directions parallel and perpendicular to the mean scalar gradient, respectively. The probability density function of the scalar flux is theoretically computed and found to be in good agreement with the numerical results. A Lagrangian statistical theory for the scalar flux is developed, which predicts that the scalar transfer flux is given by the time integral of the Lagrangian velocity autocorrelation and increases in proportion to the Péclet number, which is consistent with the results of the direct numerical simulation. A physical explanation of the asymmetry of the scalar flux PDF is explored.     

Minimum state for high Reynolds and Péclet number turbulent flows

Direct numerical simulations (DNS) or experiments for the very high Reynolds (Re) and Péclet (Pe) number flows commonly exceed the resolution possible even when use is made of the most advanced computer capability or most sophisticated diagnostics and physical capabilities of advanced laboratory facilities. In practice use is made of statistical flow data bases developed at the highest Re and Pe levels achievable within the currently available facility limitations. In addition, there is presently no metric to indicate whether and how much of the fully resolved physics of the flow of interest has been captured within the facilities available. In this Letter the authors develop the necessary metric criteria for homogeneous, isotropic and shear layer flows. It is based on establishing a smaller subset of the total range of dynamic scale interactions that will still faithfully reproduce all of the essential, significant, influences of the larger range of scale interactions. The work identifies a minimum significant Re and Pe level that must be obtained by DNS or experiment in order to capture all of the significant dynamic influences in data which is then scaleable to flows of interest. Hereafter this is called the minimum state. Determination of the minimum state is based on finding a minimum scale separation for the energy-containing scales of the flow and scalar fields sufficient to prevent contamination by interaction with the (non-universal) velocity dissipation and scalar diffusivity inertial range scale limits.     

On some transport properties of baker's maps

This paper considers the one-dimensional advection and diffusion of a passive scalar in the context of baker's maps of the unit interval. Our main interest is the thermal transport between two points held at fixed temperatures, when a deterministic sequence of maps of various scales are involved. Molecular diffusion occurs during the periods of rest between maps. We focus on the behavior of the transport in the limit of infinite Péclet number (or small molecular diffusion). Various asymptotic results are presented and compared with numerical calculations. Convergence to turbulent transport independent of molecular diffusion is observed as the number of scales is increased.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek ₀(Pe)^1/2 in the limit of large Péclet numbers wherek ₀ is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.Research partially supported by NSF DMS 90-05799 and ARO DAAL 03-89-K-0039 and AFOSR-90-0090Research partially supported by NSF DMS 87-02864, ARO DAAL 03-89-K-0013 and ONR N 00014-89-J-1044     

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Péclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet).     

Brownian motors: Joint effect of non-Gaussian noise and time asymmetric forcing

Previous works have shown that time asymmetric forcing on the one hand, as well as non-Gaussian noises on the other, can separately enhance the efficiency and current of a Brownian motor. Here, we study the result of subjecting a Brownian motor to both effects simultaneously. Our results have been compared with those obtained for the Gaussian white noise regime in the adiabatic limit. We find that, although the inclusion of the time asymmetry parameter increases the efficiency value up to a certain extent, for the present case this increase is much less appreciable than in the white noise case. We also present a comparative study of the transport coherence in the context of colored noise. Though the efficiency in some cases becomes higher for the non-Gaussian case, the Péclet number is always higher in the Gaussian colored noise case than in the white noise as well as non-Gaussian colored noise cases.     

On the influence of the hydrodynamic interactions on the aggregation rate of magnetic spheres in a dilute suspension

Magnetostatic attraction may lead to formation of aggregates in stable colloidal magnetic suspensions and magneto-rheological suspensions. The aggregation problem of magnetic composites under differential sedimentation is a key problem in the control of the instability of non-Brownian suspensions. Against these attractive forces are the electrostatic repulsion and the hydrodynamic interactions acting as stabilizing effects to the suspension. This work concerns an investigation of the pairwise interaction of magnetic particles in a dilute sedimenting suspension. We focus attention on suspensions where the Péclet number is large (negligible Brownian motion) and where the Reynolds number (negligible inertia) is small. The suspension is composed of magnetic micro-spheres of different radius and density immersed in a Newtonian fluid moving under the action of gravity. The theoretical calculations are based on direct computations of the hydrodynamic and the magnetic interactions among the rigid spheres in the regime of low particle Reynolds number. From the limiting trajectory in which aggregation occurs, we calculate the collision efficiency, representing the dimensionless rate at which aggregates are formed. The numerical results show clear evidence that the hydrodynamic interactions are of fundamental relevance in the process of magnetic particle aggregation. We compare the stabilizing effects between electrostatic repulsion and hydrodynamic interactions.     

Parametric investigation of heating due to magnetic fluid hyperthermia in a tumor with blood perfusion

Magnetic fluid hyperthermia (MFH) is a cancer treatment that can selectively elevate the tumor temperature without significantly damaging the surrounding healthy tissue. Optimal MFH design requires a fundamental parametric investigation of the heating of soft materials by magnetic fluids. We model the problem of a spherical tumor and its surrounding healthy tissue that are heated by exciting a homogeneous dispersion of magnetic nanoparticles infused only into the tumor with an external AC magnetic field. The key dimensionless parameters influencing thermotherapy are the Péclet, Fourier, and Joule numbers. Analytical solutions for transient and steady hyperthermia provide correlations between these parameters and the portions of tumor and healthy tissue that are subjected to a threshold temperature beyond which they are damaged. Increasing the ratio of the Fourier and Joule numbers also increases the tumor temperature, but doing so can damage the healthy tissue. Higher magnetic heating is required for larger Péclet numbers due to the larger convection heat loss that occurs through blood perfusion. A comparison of the model predictions with previous experimental data for MFH applied to rabbit tumors shows good agreement. The optimal MFH conditions are identified based on two indices, the fraction I_T of the tumor volume in which the local temperature is above a threshold temperature and the ratio I_N of the damaged normal tissue volume to the tumor tissue volume that also lies above it. The spatial variation in the nanoparticle concentration is also considered. A Gaussian distribution provides efficacy while minimizing the possibility of generating a tumor hot spot. Varying the thermal properties of tumor and normal tissue alters I_Tand I_N but the nature of the temperature distribution remains unchanged.     

Temporal chaos versus spatial mixing in reaction-advection-diffusion systems

We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time-independent flows and equal Pe clet numbers of different components, is demonstrated to work accurately for time-dependent flows and different Pe clet numbers.     

Diffusion and reaction-diffusion in fast cellular flows

Cellular structures, as the rolls generated by Rayleigh-Bénard instability, have always been an important topic in nonlinear science. The diffusion of a passive scalar in a given steady cellular flow becomes an interesting question in the limit of a large Péclet number, often realistic. The main result there is that the effective diffusion is somewhere in between the molecular diffusion and the "turbulent" diffusion. A new added twist to this is the reaction-diffusion case, where the front speed is the laminar propagation velocity (without flow) times the Péclet number to the power 1/4. I refine this last result and give the behavior of the prefactor in the Zel'dovich limit of a narrow reaction zone.     

Fluid mixing from viscous fingering

Mixing efficiency at low Reynolds numbers can be enhanced by exploiting hydrodynamic instabilities that induce heterogeneity and disorder in the flow. The unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid-fluid interfacial area and enhance mixing. Here we describe the dissipative structure of miscible viscous fingering, and propose a two-equation model for the scalar variance and its dissipation rate. Our analysis predicts the optimum range of viscosity contrasts that, for a given Péclet number, maximizes interfacial area and minimizes mixing time. In the spirit of turbulence modeling, the proposed two-equation model permits upscaling dissipation due to fingering at unresolved scales.     

Particle-based simulation and visualization of fluid flows through porous media

Abstract  

We propose a method of fluid simulation where boundary conditions are designed in such a way that fluid flow through porous media, pipes, and chokes can be realistically simulated. Such flows are known to be low Reynolds number incompressible flows and occur in many real life situations. To obtain a high quality fluid surface, we include a scalar value in isofunction. The scalar value indicates the relative position of each particle with respect to the fluid surface.     

The Behaviors of Ferro-Magnetic Nano-Particles In and Around Blood Vessels under Applied Magnetic Fields

In magnetic drug delivery, therapeutic magnetizable particles are typically injected into the blood stream and magnets are then used to concentrate them to disease locations. The behavior of such particles in-vivo is complex and is governed by blood convection, diffusion (in blood and in tissue), extravasation, and the applied magnetic fields. Using physical first-principles and a sophisticated vessel-membrane-tissue (VMT) numerical solver, we comprehensively analyze in detail the behavior of magnetic particles in blood vessels and surrounding tissue. For any blood vessel (of any size, depth, and blood velocity) and tissue properties, particle size and applied magnetic fields, we consider a Krogh tissue cylinder geometry and solve for the resulting spatial distribution of particles. We find that there are three prototypical behaviors (blood velocity dominated, magnetic force dominated, and boundary-layer formation) and that the type of behavior observed is uniquely determined by three non-dimensional numbers (the magnetic-Richardson number, mass Péclet number, and Renkin reduced diffusion coefficient). Plots and equations are provided to easily read out which behavior is found under which circumstances ( Fig. 5, Fig. 6, Fig. 7 and Fig. 8). We compare our results to previously published in-vitro and in-vivo magnetic drug delivery experiments. Not only do we find excellent agreement between our predictions and prior experimental observations, but we are also able to qualitatively and quantitatively explain behavior that was previously not understood.     

Anisotropic Diffusion in Driven Convection Arrays

We numerically investigate the transport of a Brownian colloidal particle in a square array of planar counter-rotating convection rolls at high Péclet numbers. We show that an external force produces huge excess peaks of the particle’s diffusion constant with a height that depends on the force orientation and intensity. In sharp contrast, the particle’s mobility is isotropic and force independent. We relate such a nonlinear response of the system to the advection properties of the laminar flow in the suspension fluid.     

On the nonlinear dynamics of a saline boundary layer formed by throughflow near the surface of a porous medium

We consider gravitational instability of saline boundary layers, observed at the subsurface of salt lakes. This boundary layer is the result of the convective transport induced by the evaporation at the horizontal surface of a confined porous medium. When this upward transport is balanced by salt dispersion, a steady state boundary layer is formed. However, this boundary layer can be unstable when perturbed. This results in complex groundwater motion and density fields. The aim of this paper is to investigate the existence of finite amplitude solutions describing these resulting patterns (both the number of solutions and their structure), their stability, and their dependency on the system Rayleigh and Péclet numbers. For this purpose we construct a low-dimensional dynamical system (a reduced model) by projecting the nonlinear model equations onto a relatively small set of eigenfunctions of the problem linearized at criticality. The Galerkin projection approach is complicated by the fact that the problem under consideration is non-self-adjoint due to the existing evaporation. This implies that the eigenfunctions do not form an orthogonal set and therefore the adjoint eigenfunctions are used for the projection. The reduced model is constructed in such a way that it is capable of providing solutions in the strongly nonlinear regime as well. Convergence of these solutions towards the fully nonlinear model results is shown by means of direct numerical simulations. Further, the reduced model seems to partly capture the complex nonlinear behavior as seen in Hele-Shaw experiments by Wooding et al. [R.A. Wooding, S.W. Tyler, I. White, P.A. Anderson, Convection in groundwater below an evaporating salt lake: 2. evolution of fingers or plumes, Water Resour. Res. 33 (6) (1997) 1219-1228]. The physical transition mechanism that explains the occurrence of some observed bifurcation types is presented as well.     

An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion

The fundamental soundness of three flamelet models for non-premixed turbulent combustion is examined on the basis of their performance in an idealized model problem that merges ideas from the laminar asymptotic theory for non-premixed flames and rigorous homogenization theory for the diffusion of a passive scalar. The overall flame configuration is stabilized by a mean gradient in the passive scalar: large Damköhler number asymptotics results are available for the laminar case to quantify the finite-rate effects that cause the flame to depart from its equilibrium state; the same results can also be used to incorporate higher-order corrections in the approximation of the reactive variables in terms of the passive scalar. The use of such flamelet approximations has been extended well beyond the laminar regime as they lie at the core of practical strategies to simulate non-premixed flames in the turbulent regime: the flamelet representation avoids the problem of turbulence closure for the reactive variables by replacing it by the presumably much simpler closure problem for a passive scalar. It is precisely the validity of this substitution outside the laminar regime that is addressed here in the idealized context of a class of small-scale periodic flows for which extensive rigorous results are available for the passive scalar statistics. Results for this simplified problem are reported here for significant wide ranges of Peclet and Damköhler numbers. Asymptotic convergence is observed in terms of the Damköhler number, with a convergence rate that is found to match the laminar predictions and appears relatively insensitive to the Peclet number. The passive scalar dissipation plays a key role in achieving higher-order corrections for the finite-rate case: replacing its pointwise value by an averaged value is convenient practically and can be rigorously motivated for the class of flows studied here, but while it does achieve an overall improvement over the lower-order equilibrium model, the simplification compromises the higher asymptotic convergence observed with the original finite-rate flamelet model with exact local dissipation.(Some figures in this article are in colour only in the electronic version; see www.iop.org)     

Nature's microfluidic transporter: rotational cytoplasmic streaming at high Péclet numbers

Cytoplasmic streaming circulates the contents of large eukaryotic cells, often with complex flow geometries. A largely unanswered question is the significance of these flows for molecular transport and mixing. Motivated by "rotational streaming" in Characean algae, we solve the advection-diffusion dynamics of flow in a cylinder with bidirectional helical forcing at the wall. A circulatory flow transverse to the cylinder's long axis, akin to Dean vortices at finite Reynolds numbers, arises from the chiral geometry. Strongly enhanced lateral transport and longitudinal homogenization occur if the transverse Péclet number is sufficiently large, with scaling laws arising from boundary layers.     

New version of seven wavelengths demultiplexer based on the microcavities in a two-dimensional photonic crystal

A new two dimensional photonic crystal demultiplexer of wavelength (WDM) is designed by exploiting two Fabry–Pérot reflectors at the end of the bus waveguides. The results show that the light with different wavelengths can be successfully filtered to different ports by setting different radius of the center defect rods in the drop waveguides and high drop efficiency can be achieved by means of reflection feedbacks. The proposed filter has a cross section equal to 9.7 μm × 5.8 μm. In the designed filter, an improvement of the number of channels has been achieved. The normalized transmission spectra of this component have been studied using finite difference time domain (FDTD) method. The important parameters consider for this studies are radius of rods used in Fabry–Pérot reflectors, and radius of center defect rods in the drop waveguides. The demultiplexer we designed can easily separate the light with seven different wavelengths simultaneously. The scope of this paper lies on demultiplexer for communication systems around 1.55-μm wavelength.     

Fermion states on domain-wall junctions and the flavor number

In this paper we address the problem of localizing fermion states on stable domain-wall junctions. The study focus on the consequences of intersecting six independent 8d domain walls to form 4d junctions in a ten-dimensional spacetime. This is related to the mechanism of relaxing to three space dimensions through the formation of domain-wall junctions. The model is based on six bulk real scalar fields, the φ ⁴ model in its broken phase, the prototype of the Higgs field, and is such that the fermion and scalar modes bound to the domain walls are the zero mode and a single massive bound state, which can be regarded as a two-level system, at least at sufficiently low energy. Inside the junction, we use the fact that some states are statistically more favored to address the possibility of constraining the flavor number of the elementary fermions.