Passive scalar transport in peripheral regions of random flows

We investigate statistical properties of the passive scalar mixing in random (turbulent) flows assuming its diffusion to be weak. Then at advanced stages of the passive scalar decay, its unmixed residue is primarily concentrated in a narrow diffusive layer near the wall and its transport to the bulk goes through the peripheral region (laminar sublayer of the flow). We conducted Lagrangian numerical simulations of the process for different space dimensions d and revealed structures responsible for the transport, which are passive scalar tongues pulled from the diffusive boundary layer to the bulk. We investigated statistical properties of the passive scalar and of the passive scalar integrated along the wall. Moments of both objects demonstrate scaling behavior outside the diffusive boundary layer. We propose an analytic scheme for the passive scalar statistics, explaining the features observed numerically.     

Passive scalar intermittency in low temperature helium flows

We report new measurements of mixing of passive temperature field in a turbulent flow. The use of low temperature helium gas allows us to span a range of microscale Reynolds number, R(lambda), from 100 to 650. The exponents xi(n) of the temperature structure functions approximately r(xi(n)) are shown to saturate to xi(infinity) approximately 1.45+/-0.1 for the highest orders, n approximately 10. This saturation is a signature of statistics dominated by frontlike structures, the cliffs. Statistics of the cliffs' characteristics are performed, particularly their widths are shown to scale as the Kolmogorov length scale.     

Passive scalar in a random wave field: the weak turbulence approach

We consider the evolution of a passive scalar advected by a velocity field which is a superposition of random linear waves. An equation for the average concentration of the passive scalar is derived (in the limit of small molecular diffusion) using the weak turbulence methodology. In addition to the enhanced diffusion, this equation contains the correction to the (Stokes) drift. Both of these terms have the fourth order with respect to wave amplitudes. The formulas for the coefficients of turbulent diffusion and turbulent drift are derived.     

Passive scalar in a large-scale velocity field

We consider advection of a passive scalar θ(t,r) by an incompressible large-scale turbulent flow. In the framework of the Kraichnan model all PDF’s (probability distribution functions) for the single-point statistics of θ and for the passive scalar difference θ(r ₁)−θ(r ₂) (for separations r ₁−r ₂ lying in the convective interval) are found. Zh. éksp. Teor. Fiz. 115, 920–939 (March 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Kr-PLIF for scalar imaging in supersonic flows

Experiments were performed to explore the use of two-photon planar laser-induced fluorescence (PLIF) of krypton gas for applications of scalar imaging in supersonic flows. Experiments were performed in an underexpanded jet of krypton, which exhibited a wide range of conditions, from subsonic to hypersonic. Excellent signal-to-noise ratios were obtained, showing the technique is suitable for single-shot imaging. The data were used to infer the distribution of gas density and temperature by correcting the fluorescence signal for quenching effects and using isentropic relations. The centerline variation of the density and temperature from the experiments agree very well with those predicted with an empirical correlation and a CFD simulation (FLUENT). Overall, the high signal levels and quantifiable measurements indicate that Kr-PLIF could be an effective scalar marker for use in supersonic and hypersonic flow applications.     

Reactive particles in random flows

We study the dynamics of chemically or biologically active particles advected by open flows of chaotic time dependence, which can be modeled by a random time dependence of the parameters on a stroboscopic map. We develop a general theory for reactions in such random flows, and derive the reaction equation for this case. We show that there is a singular enhancement of the reaction in random flows, and this enhancement is increased as compared to the nonrandom case. We verify our theory in a model flow generated by four point vortices moving chaotically.     

Convection-enhanced diffusion for random flows

We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of Isichenko and Kalda that the effective diffusivity behaves likeɛ ^3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows.     

Dynamical slowdown of polymers in laminar and random flows

The influence of an external flow on the relaxation dynamics of a single polymer is investigated theoretically and numerically. We show that a pronounced dynamical slowdown occurs in the vicinity of the coil-stretch transition, especially when the dependence on polymer conformation of the drag is accounted for. For the elongational flow, relaxation times are exceedingly larger than the Zimm relaxation time, resulting in the observation of conformation hysteresis. For random smooth flows, hysteresis is not present. Yet, relaxation dynamics is significantly slowed down because of the large variety of accessible polymer configurations. The implications of these results for the modeling of dilute polymer solutions in turbulent flows are addressed.     

Merger of coherent structures in time-periodic viscous flows

Inertia-induced changes in transport properties of an incompressible viscous time-periodic flow are studied in terms of the topological properties of volume-preserving maps. In the noninertial limit, the flow admits one constant of motion and thus relates to a so-called one-action map. However, the invariant surfaces corresponding to the constant of motion are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the effect of inertia and leads to a new kind of response scenario: resonance-induced merger of coherent structures.     

Diffusion in layered random flows,polymers, electrons in random potentials,and spin depolarization in random fields

Several related models are studied in a common framework. We first reconsider the model of Matheron and de Marsilly for (anomalous) tracer dispersion in a stratified porous medium. In each horizontal layer the flow velocity is constant, parallel to the layer, and depends randomly on the vertical coordinate z. This model is mapped onto ad=1 localization problem in a random potential and, equivalently, onto ad=1 polymer. At larget theaveraged distribution of horizontal displacementsx takes the scaling form [P(x, t, z=0)]=at ^–5/4 Q(bxt ^–3/4), whereQ(y) is independent of the details of the model.Q(y),a, andb are obtained exactly for a large class of models. From the Lifschitz tails of the localization problem we find in the regionxt ^3/4, i.e.,y, thatQ(y)¦y¦ exp(–C¦y¦^4/3). We also obtain exactly ind=1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field, by mapping onto a polymer problem, as well as the average local concentration for diffusion in the presence of random sources and sinks. These mappings are then used to study higher-dimensional extensions of these models.     

Percolation transport in random flows with weak dissipation effects

In the present paper, we consider the influence of weak dissipative effects on the passive scalar behavior in the framework of continuum percolation approach. The renormalization method of a small parameter in continuum percolation models is reviewed. It is shown that there is a characteristic velocity scale, which corresponds to the dissipative process. The modification of the renormalization condition of the small percolation parameter is suggested in accordance with additional external influences superimposed on the system. In the framework of mean-field arguments, the balance of correlation scales is considered. This gives the characteristic time that corresponds to the percolation regime. The expression for the effective coefficient of diffusion is obtained.     

Generalized Joseph estimates of stability of plane shear flows with scalar nonlinearity

Possible statements of eigenvalue problems generalizing the classical Orr-Sommerfeld problem are given for incompressible stationary flows of non-Newtonian fluids; these problems are interpreted within the mechanics of a continuum as problems of the shear stability of such flows. A schematic diagram of the integral relation method as applied to these flows is described. In the case of an unperturbed Couette flow, Joseph estimates are generalized for any type of rheological curve and domains of guaranteed stability are constructed on a plane with the axes showing Reynolds numbers corresponding to the tangent and secant dynamic viscosity at one point.     

Diffusion and clustering of sedimenting tracers in random hydrodynamic flows

The analysis presented in [1, 2] is extended to sedimenting low-inertia tracers advected by random divergence-free hydrodynamic flows. The key feature of the process is the clustering of the tracers due to the divergence of tracer-velocity field. This phenomenon has probability one; i.e., it takes place in almost every realization of the process. Both spatial diffusivity and diffusivity in the density space (responsible for clustering) are calculated. The low inertia of the tracers does not affect the spatial diffusivity. The indispensable use of a finite velocity correlation time leads to an anisotropic spatial diffusivity. The calculations performed in the study are based on a diffusion approximation.     

Complex structures adapted to magnetic flows

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g$, and let $\beta$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^{1}M$ that describes a charged particle moving in the magnetic field $\beta$. Following an idea of T. Thiemann, we construct a complex structure on a tube inside $T^{1}M$ by pushing forward the vertical polarization by the Hamiltonian flow “evaluated at time $i$”. This complex structure fits together with $\omega -{\pi }^1\beta$ to give a Kähler structure on a tube inside $T^{1}M$. When $\beta =0$, our magnetic complex structure is the adapted complex structure of Lempert–Szőke and Guillemin–Stenzel.We describe the magnetic complex structure in terms of its $(1,0)$-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney–Bruhat and Grauert. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by $-\beta$, and we give two formulas for local Kähler potentials, which depend on a local choice of vector potential $1$-form for $\beta$. Finally, we compute the magnetic complex structure explicitly for constant magnetic fields on ${\mathbb{R}}^2$ and $S^2$.     

Hamiltonian and gradient structures in the Toda flows

In this paper we consider gradient structures in the dynamics and geometry of the asymmetri nonperiodic tridiagonal and full Toda flow equations. We compare and contrast a number of formulations of the nonperiodic Toda equations. In the case of the full Kostant (asymmetric) Toda flow we explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and prove in particular that one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map.     

The mechanics of particle accumulation structures in thermocapillary flows

The motion of small particles suspended in a cylindrical thermocapillary liquid bridge is considered. Owing to geometry and surface stresses the streamlines gather near the cylindrical free surface and provoke particle–free-surface collisions. We show numerically that tracers which are perfect but of finite size can accumulate on closed trajectories. A simple model is proposed to explain the attraction of particles to the closed trajectory based on the flow topology in the vicinity of a closed streamline which comes sufficiently close to the free surface and on particle–free-surface collisions which transfer particles among different streamlines.