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We introduce a class of multi-scale systems with discrete time, motivated by the problem of inviscid limit in fluid dynamics in the presence of small-scale noise. These systems are infinite-dimensional and defined on a scale-invariant space-time lattice. We propose a qualitative theory describing the vanishing regularization (inviscid) limit as an attractor of the renormalization group operator acting in the space of flow maps or respective probability kernels. If the attractor is a nontrivial probability kernel, we say that the inviscid limit is spontaneously stochastic: it defines a stochastic (Markov) process solving deterministic equations with deterministic initial and boundary conditions. The results are illustrated with solvable models: symbolic systems leading to digital turbulence and systems of expanding interacting phases. [ABSTRACT FROM AUTHOR]

We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence. [ABSTRACT FROM AUTHOR]

The butterfly effect is today commonly identified with the sensitive dependence of deterministic chaotic systems upon initial conditions. However, this is only one facet of the notion of unpredictability pioneered by Lorenz, who actually predicted that multiscale fluid flows could spontaneously lose their deterministic nature and become intrinsically random. This effect, which is radically different from chaos, have remained out of reach for detailed physical observations. Here we show that this scenario is inherent to the elementary Kelvin–Helmholtz hydrodynamical instability of an initially singular shear layer. We moreover provide evidence that the resulting macroscopic flow displays universal statistical properties that are triggered by, but independent of specific physical properties at micro-scales. This spontaneous stochasticity is interpreted as an Eulerian counterpart to Richardson's relative dispersion of Lagrangian particles, giving substance to the intrinsic nature of randomness in turbulence. Intrinsic randomness, also known as spontaneous stochasticity, has been suggested to limit the finite-time predictability of multiscale chaotic dynamics, but an explicit demonstration of its effect is still lacking. Here, the authors use numerical simulations to show the existence of spontaneous stochasticity in the discontinuous shear layer leading to the Kelvin-Helmholtz instability. [ABSTRACT FROM AUTHOR]

Volatile oil recovery by means of air injection is studied as a method to improve recovery from low permeable reservoirs. We consider the case in which the oil is directly combusted into small products, for which we use the term medium temperature oil combustion. The two-phase model considers evaporation, condensation and reaction with oxygen. In the absence of thermal, molecular and capillary diffusion, the relevant transport equations can be solved analytically. The solution consists of three waves, i.e., a thermal wave, a medium temperature oxidation (MTO) wave and a saturation wave separated by constant state regions. A striking feature is that evaporation occurs upstream of the combustion reaction in the MTO wave. The purpose of this paper is to show the effect of diffusion mechanisms on the MTO process. We used a finite element package (COMSOL) to obtain a numerical solution; the package uses fifth-order Lagrangian base functions, combined with a central difference scheme. This makes it possible to model situations at realistic diffusion coefficients. The qualitative behavior of the numerical solution is similar to the analytical solution. Molecular diffusion lowers the temperature of the MTO wave, but creates a small peak near the vaporization region. The effect of thermal diffusion smoothes the thermal wave and widens the MTO region. Capillary diffusion increases the temperature in the upstream part of the MTO region and decreases the efficiency of oil recovery. At increasing capillary diffusion the recovery by gas displacement gradually becomes higher, leaving less oil to be recovered by combustion. Consequently, the analytical solution with no diffusion and numerical solutions at a high capillary diffusion coefficient become different. Therefore high numerical diffusion, significant in numerical simulations especially in coarse gridded simulations, may conceal the importance of combustion in recovering oil. [ABSTRACT FROM AUTHOR]

There is a renewed interest in using combustion for the recovery of medium viscosity oil. We consider the combustion process when air is injected into the porous medium containing some fuel and inert gas. Commonly the reaction rate is negligible at low temperatures, hence the possibility of oxygen breakthrough. In this case, the oxygen gets in contact with the fuel in the downstream zone leading to slow reaction. We focus on the case when the reaction is active for all temperatures, but heat losses are negligible. For a combustion model that includes heat and mass balance equations, we develop a method for calculating the wave profile in the form of an asymptotic expansion and derive its zero- and first-order approximations. This wave profile appears to be different from wave profiles analyzed in other papers, where only the reaction at the highest temperatures was taken into account. The combustion wave has a long decaying tail. This tail is hard to observe in the laboratory because heat losses must be very small for the long tail to form. Numerical simulations were performed in order to validate our asymptotic formulae. [ABSTRACT FROM AUTHOR]

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler's paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form. [ABSTRACT FROM AUTHOR]

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S i, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S i, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S i, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S i, j shocks and classical waves are described. As a physical application, all types of S i, j shocks with i> j are detected and studied in a family of models for multi-phase flow in porous media. [ABSTRACT FROM AUTHOR]