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  • 1
    Electronic Resource
    Electronic Resource
    Enhanced diffusivity and intercell transition layers in 2-D models of passive advection (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 3209-3212 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A class of two-dimensional, isotropic, divergence-free vector fields is introduced and the effective diffusivity of the corresponding advection–diffusion equations is studied. These examples are very idealized flows, but they can be solved exactly in the limit Pe(very-much-greater-than)1. Scaling laws D*∝D0(Pe)α are obtained, where D0=molecular diffusion, Pe=Peclet number, with exponents in the range 0〈α〈1, and examples of "stream functions'' with logarithmic singularities for which D*∝D0Pe. The exponent α is related by a simple formula to the shape of the stream function along cell boundaries, suggesting that similar scaling laws should hold for more general 2-D closed-cell flows.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529480
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  • 2
    Electronic Resource
    Electronic Resource
    On Woltjer's variational principle for force-free fields (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 1240-1253 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The existence of minimizers for Woltjer's variational principle is established and that the minimizers are force-free fields. This method has the nature of a constructive implicit function theorem and handles successfully the nonconvex constraint of constant total helicity. Domains of arbitrary connectivity are allowed as well as nonhomogeneous boundary conditions and periods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529321
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  • 3
    Electronic Resource
    Electronic Resource
    Effective dielectric and elastic constants of piezoelectric polycrystals (1992)
    [S.l.] : American Institute of Physics (AIP)
    Journal of Applied Physics 71 (1992), S. 4455-4464 
    Details
    ISSN: 1089-7550
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The effective dielectric constant, bulk modulus, and shear modulus of isotropic polycrystals with piezoelectric grains are studied using an effective medium approximation (EMA) and generalized Hashin–Shtrikman bounds. The EMA determines self-consistently the electromechanical interaction of grains with the surrounding composite. Numerical values for the moduli are computed for barium titanate and compared with available experimental data, as well as with classical estimates for the moduli. Further assessment of the EMA is made by computing numerical values of the effective moduli for ideal polycrystals, based on numerical data for crystals with strong piezoelectric coupling and comparing the resulting values with classical estimates. Similar comparisons are made for the generalized Hashin–Shtrikman bounds. On "ideal'' polycrystals the gap between the upper and lower bounds can be 30% narrower than the corresponding gap if piezoelectric coupling is neglected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.350788
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  • 4
    Electronic Resource
    Electronic Resource
    Approximate and exact renormalization theories for a model for turbulent transport (1992)
    New York, NY : American Institute of Physics (AIP)
    Physics of Fluids 4 (1992), S. 41-57 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The capability of renormalization group methods (RNG) and Lagrangian renormalized perturbation theories (RPT) to reproduce a renormalized theory of eddy diffusivity for turbulent transport diffusion is discussed in the context of a simplified model with an exact renormalization theory that has recently been developed by the authors. The model problem involves transport diffusion by simple shear flows with turbulent velocity statistics, infrared divergences, and no separation of scales; the exact renormalization theory exhibits a remarkable range of different phenomena as parameters defining the velocity statistics are varied with four distinct regions requiring renormalization so that the model is a stringent test for approximate theories of eddy diffusivity via either RNG or RPT methods. Despite the different philosophy in RNG and RPT methods, all of these different approximations collapse to give the exact theory of eddy diffusivity for one region in the model with infrared divergence that is adjacent to the Kolmogorov value. The RNG methods are very flexible but do not give the exact anomalous scaling exponents for the other three regions with infrared divergence as expected with an ε-expansion procedure. The Lagrangian RPT methods always yield the correct scaling exponents but a much more elaborate analysis of the explicit structure of the model problem is needed to achieve this. In other regions of renormalization, including the Kolmogorov value, the RPT methods predict nonlocal equations for eddy diffusivity while the exact renormalization theory involves local diffusion equations with time-dependent diffusivity; these nonlocal equations are a poor approximation for the actual renormalized dynamics and the Lagrangian direct interaction approximation (DIA) only slightly improves the behavior over the Lagrangian first-order smoothing approximation. On the other hand, RNG methods alway predict a simple local diffusivity in the model and there are regions of renormalization where the rigorous theory for eddy diffusivity is nonlocal.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.858499
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  • 5
    Electronic Resource
    Electronic Resource
    Effective conductivity and average polarizability of random polycrystals (1990)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2047-2056 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A third-order expansion for the effective thermal conductivity tensor @sK* of anisotropic polycrystalline cell materials is derived. The coefficients of the expansion are given in terms of the average polarizability tensor, a nondimensional quantity determined from the grain shape and crystallographic orientation distributions independent of other details of the microgeometry such as two (or more) particle correlation functions. Explicit numerical results for a wide variety of microgeometries made of ellipsoidal cells are obtained. This calculation uses a new method that exploits the symmetry properties of the effective conductivity tensor of a cell material as a function of the single-crystal conductivities.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528656
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  • 6
    Electronic Resource
    Electronic Resource
    An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows (1991)
    Springer
    Communications in mathematical physics 138 (1991), S. 339-391 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099496
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  • 7
    Electronic Resource
    Electronic Resource
    Mathematical models with exact renormalization for turbulent transport (1990)
    Springer
    Communications in mathematical physics 131 (1990), S. 381-429 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The advection-diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and/or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media. A variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts. In this paper, a simple model problem for the advection-diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor. Explicit formulas for the anomalous time scaling in various regimes as well as the Green's function for the large-scale, long-time, ensemble average are developed here. Formulas for the renormalized higher order statistics are also developed. The simple form of the model problem is deceptive; the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied. These phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter $$\tilde \varepsilon $$ is varied as well as explicit regimes in $$\tilde \varepsilon $$ where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity. We use Fourier analysis and the Feynman-Kac formula as main tools in the explicit exact renormalization theory developed here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02161420
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  • 8
    Electronic Resource
    Electronic Resource
    Statistical properties of shocks in Burgers turbulence, II: Tail probabilities for velocities, shock-strengths and rarefaction intervals (1995)
    Springer
    Communications in mathematical physics 169 (1995), S. 45-59 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This paper studies the structure of the random “sawtooth” profile corresponding to the solution of the inviscid Burgers equation with white-noise initial data. This function consists of a countable sequence of rarefaction waves separated by shocks. We are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity, shock-strength and rarefaction intervals. We find that these quantities have tail probabilities of the form exp{−Cx 3},x≫1. This “cubic exponential” decay of probabilities was conjectured in the companion paper [1]. The calculations are done using a representation of the shock-strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02101596
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  • 9
    Electronic Resource
    Electronic Resource
    Statistical properties of shocks in Burgers turbulence (1995)
    Springer
    Communications in mathematical physics 172 (1995), S. 13-38 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider the statistical properties of solutions of Burgers' equation in the limit of vanishing viscosity, $$\frac{\partial }{{\partial t}}u\left( {x,t} \right) + \frac{\partial }{{\partial x}}\left( {\frac{1}{2}u\left( {x,t} \right)^2 } \right) = 0$$ , with Gaussian whitenoise initial data. This system was originally proposed by Burgers[1] as a crude model of hydrodynamic turbulence, and more recently by Zel'dovichet al..[12] to describe the evolution of gravitational matter at large spatio-temporal scales, with shocks playing the role of mass clusters. We present here a rigorous proof of the scaling relationP(s)∞s 1/2,s≪1 whereP(s) is the cumulative probability distribution of shock strengths. We also show that the set of spatial locations of shocks is discrete, i.e. has no accumulation points; and establish an upper bound on the tails of the shock-strength distribution, namely 1−P(s)≤exp{−Cs 3} fors≫1. Our method draws on a remarkable connection existing between the structure of Burgers turbulence and classical probabilistic work on the convex envelope of Brownian motion and related diffusion processes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02104509
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  • 10
    Electronic Resource
    Electronic Resource
    Mathematical models with exact renormalization for turbulent transport, II: Fractal interfaces, non-Gaussian statistics and the sweeping effect (1992)
    Springer
    Communications in mathematical physics 146 (1992), S. 139-204 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors. Three important topics are analyzed with complete mathematical rigor for this model: (1) Renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces, (2) the effect of non-Gaussian turbulent velocity statistics on renormalization theory, (3) the “sweeping” effect of additional large scale mean velocities. A special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the Kolmogorov-spectrum in the model. In the authors' earlier paper, it was established that the Kolmogorov value is at a phase transition boundary in the exact renormalization theory. It is found here that the qualitative model, despite its simplicity contains, in the vicinity of the Kolmogorov value, a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories. In particular, the Richardson 4/3-law for pair dispersion and interfaces with fractal dimension defect of 2/3 occur in the model rigorously as limits when the Kolmogorov spectrum is approached as a limit from one side of the phase transition boundary; alternative corrections to the Richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1/3, occur in the model when the Kolmogorov spectrum is approached from the other side of the phase transition. It is very interesting that fractal dimension defects of roughly the value either 1/3 or 2/3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments. As regards non-Gaussian the asymptotic normality of normalized integrals (B.56) corresponding to compactly supported blobs with mean zero. The proof of this latter fact is done in the same way as Step 2, Proposition B.3, using the fact that the corresponding random processes $$\tilde V_\delta (s)$$ have finite domain of dependence. This concludes the proof of Proposition B.4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099212
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