Transition to self-similarity of diffusion of tracer in turbulent patch

[Abstract]: A mixing of a passive tracer inside a turbulent patch generated by a localized short-time perturbation is studied numerically and analytically. Two kinds of an initial distribution of a tracer are considered: two-layer and continuous with constant gradient. For the turbulent patch shaped as a layer, it is shown that, regardless of details of initial distributions of a turbulent energy and dissipation, a tracer concentration evolves to self-similar regimes as time elapses. Analytical self-similar solutions to turbulent diffusion equations are found for three symmetric shapes of a turbulent patch: layer, cylinder, and sphere. Distributions of the concentration inside a patch are found to be substantially nonuniform, with a typical ratio of a concentration gradient in the middle of a patch to its initial value of about 0.5

The role of rheology in modelling elastic waves with gas bubbles in granular fluid-saturated media

Elastic waves in fluid-saturated granular media depend on the rheology which includes elements representing the fluid and, if necessary, gas bubbles. We investigated the effect of different rheological schemes, including and excluding the bubbles, on the linear Frenkel–Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum, and a bubble surrounded by the fluid. We derived the Nikolaevskiy-type equations describing the velocity of the solid matrix in the moving reference system. The equations are linearized to yield the decay rate lambda as a function of the wave number k. We compared the lambda(k)-dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For the both cases, the lambda(k)-curve lies entirely below zero, which is in line with the notion of the elastic wave being an essentially passive system. We found that the increase of the radius of the bubbles leads to faster decay, while the increase in the number of the bubbles leads to slower decay of the elastic wave

Rheology and decay rate for Frenkel-Biot P1 waves in porous media with gas bubbles

We study the effect of using different rheological schemes in models of Frenkel-Biot elastic waves of P-type in porous media. Two rheological schemes are considered -one with the bubbles and the other without. The bubble-including scheme consists of segments representing the solid continuum and bubbles inside the fluid, while the bubble-free scheme is represented by the standard solid-fluid rheological model. We derived the dispersion relations for the wave equations in their linear forms and analyzed the decay rate, lambda, versus the wave number, k. We compared the lambda(k)-dependence for the two rheologies under consideration using typical values of the mechanical parameters of the model. We observed, in particular, that an increase of the radius and the number of the bubbles leads to an increase in the decay rate

Using 1D-IRBFN method for solving high-order nonlinear differential equations arising in models of active-dissipative systems

We analyse a nonlinear partial differential equation modelling reaction-diffusion systems with nonlocal coupling and reaction fronts of gasless combustion. The equation is of active-dissipative type, nonlinear, with 6th-order spatial derivative. To numerically solve the equations we use the one-dimensional integrated radial basis function network (1D-IRBFN)method. The method has been previously developed and successfully applied to several problems such as structural analysis, viscous and viscoelastic flows and fluid-structure interaction. A commonly used approach is to differentiate a function of interest to obtain approximate derivatives. However, this leads to a reduction in convergence rate for derivatives and this reduction is an increasing function of derivative order. Accordingly, differentiation magnifies errors. To avoid this problem and recognising that integration is a smoothing process, the proposed 1D-IRBFN method uses the integral formulation, where spectral approximants are utilised to represent highest-order derivatives under consideration and then integrated analytically to yield approximate expressions for lower-order derivatives and the function itself. Our preliminary results demonstrate good performance of the 1D-IRBFN algorithm for the equation under consideration. Numerical solutions representing travelling waves are obtained, in agreement with the earlier studies

Simulations of autonomous fluid pulses between active elastic walls using the 1D-IRBFN Method

We present numerical solutions of the semi-empirical model of self-propagating fluid pulses (auto-pulses) through the channel simulating an artificial artery. The key mechanism behind the model is the active motion of the walls in line with the earlier model of Roberts. Our model is autonomous, nonlinear and is based on the partial differential equation describing the displacement of the wall in time and along the channel. A theoretical plane configuration is adopted for the walls at rest. For solving the equation we used the One-dimensional Integrated Radial Basis Function Network (1D-IRBFN) method. We demonstrated that different initial conditions always lead to the settling of pulse trains where an individual pulse has certain speed and amplitude controlled by the governing equation. A variety of pulse solutions is obtained using homogeneous and periodic boundary conditions. The dynamics of one, two, and three pulses per period are explored. The fluid mass flux due to the pulses is calculated

A new case of truncated phase equation for coupled oscillators

[Abstract]: Generalized nonlinear phase diffusion equation describes oscillators weakly coupled by diffusion. The equation generally contains infinite number of terms and allows a variety of dynamic balances between them. We consider a truncated version of the equation in which nonlinear excitation drives the dynamics. A group of active systems leading to this truncation is modelled by reaction-diffusion equations with effective nonlocal coupling. We formulate the conditions on the parameters resulting in the truncation and discuss numerical experiments showing complex spatio-temporal behaviour

Attractors in confined source problems for coupled nonlinear diffusion

In processes driven by nonlinear diffusion, a signal from a concentrated source is confined in a finite region. Such solutions can be sought in the form of power series in a spatial coordinate. We use this approach in problems involving coupled agents. To test the method, we consider a single equation with (a) linear and (b) quadratic diffusivity in order to recover the known results. The original set of PDEs is converted into a dynamical system with respect to the time-dependent series coefficients. As an application we consider an expansion of a free turbulent jet. Some example trajectories from the respective dynamical system are presented. The structure of the system hints at the existence of an attracting center manifold. The attractor is explicitly found for a reduced version of the system

On characteristic times in generalized thermoelasticity

The model of Green and Lindsay is a popular generalization of the theory of thermoelasticity incorporating second sound. Within the model the second sound is intimately linked to a presence of two characteristic times, t1 and t2, constrained by an inequality t2<=t1. We present a modification of the theory where no constraints on the times arise

Phenomenological approach to 3D spinning combustion waves: numerical experiments with a rectangular rod

The spinning waves occur in solid flames and detonation when the plane uniformly propagating reaction front loses stability. As a result, the front breaks into localized zones of intensive reaction. We study a 3D phenomenological model aimed at modeling such phenomena. The model constitutes a nonlinear partial differential equation. This work contains preliminary results demonstrating the capacity of the model to reproduce basic experimental features of the unstable front: metastability of the uniform state and formation of self-sustained regime with predominantly lateral propagation of the front curvature