Well-posedness for compressible Rayleigh-Bénard convection

The Rayleigh-B~nard convection is a classical problem in fluid dynamics. In this paper, we are concerned with the well-posedness for the compressible Rayleigh-B~nard convection in a bounded domain Ω R2. We prove the local well-posedness of the system with appropriate initial data. This is the result concerning compressible Rayleigh-B~nard convection, before only results about incompressible Rayleigh-B~nard convection were done.     

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

In this paper, the initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating solution with expansions up to any order are given and the convergence rates O(ɛ ^m+1/2) and the optimal convergence rates O(ɛ ^m+1) are obtained respectively. This improves the result of J.G. SHI.     

Numerical analysis of a strongly coupled system of two singularly perturbed convection–diffusion problems

A system of two coupled singularly perturbed convection–diffusion ordinary differential equations is examined. The diffusion term in each equation is multiplied by a small parameter, and the equations are coupled through their convective terms. The problem does not satisfy a conventional maximum principle. Its solution is decomposed into regular and layer components. Bounds on the derivatives of these components are established that show explicitly their dependence on the small parameter. A numerical method consisting of simple upwinding and an appropriate piecewise-uniform Shishkin mesh is shown to generate numerical approximations that are essentially first order convergent, uniformly in the small parameter, to the true solution in the discrete maximum norm.      

Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process

The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Lévy process. One of the problem when we use Lévy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Lévy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs.     

Numerical simulation of crack–propagation in shells

We develop a finite element method for the simulation of fragmentation of thin shells. The method is valid for completely non–linear problems, but is restricted to through–the–thickness cracks, which are normal to the midsurface. The methodology is based on the extended finite element method (X–FEM). The use of X–FEM allows arbitrary crack path evolution and does not require a priori knowledge of the crack zone or remeshing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of Richtmyer-Meshkov instability

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock     

The effect of suspended particles on Rayleigh-Bénard convection I. A nonlinear stability analysis of a thermal equilibrium model

The weakly nonlinear stability of the pure conduction solution for an appropriate aerosol one-layer Rayleigh-Bénard model of a Boussinesq particle-gas system in thermal equilibrium which retains both the particle and collision pressures is investigated. The main result of this analysis is in qualitative accord with the dominant but heretofore anomalous characteristic of columnar instabilities observed in smoke-air mixtures: namely, that lowering the threshold temperature gradient associated with the occurrence of the supercritically equilibrated rolls predicted for a clean gas leads to reduction increasing with decreasing layer depth which becomes quite severe in the case of very thin layers.     

The effect of suspended particles on Rayleigh-Bénard convection II. A nonlinear stability analysis of a thermal disequilibrium model

The weakly nonlinear stability of the pure conduction solution for an appropriate aerosol one-layer Rayleigh-Bénard model of a Boussinesq particle-gas system retaining both the particle and collision pressures and considering particle to particle radiative effects while relaxing the usual assumption of thermal equilibrium between those particles and the gas is investigated. Then an analysis of the criteria governing the occurrence of supercritically re-equilibrated stationary rolls yields a minimum Rayleigh number and a critical wavelength which are completely compatible in their layer-depth behavior with normal convective and columnar instabilities observed in mixtures of smoke with air or carbon dioxide.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Simultaneous identification of convection velocity and source strength in a convection–diffusion equation

In this work we are concerned with the analysis on a simultaneous finite element reconstruction of the convection velocity and source strength in a time-dependent convection–diffusion equation. The ill-posed problem is formulated into an output least-squares nonlinear minimization by an appropriately selected Tikhonov regularization. The regularizing effect and mathematical properties of the regularized system are justified and demonstrated. The nonlinear optimization problem is approximated by a fully discrete finite element method, whose convergence is rigorously established.     

Mathematical modeling of thermal convection in a 3D model of the “DAKON” convection sensor

A three-dimensional time-dependent mathematical model of thermal convection in a cubic convection sensor is developed. An efficient numerical algorithm is designed for computing convective flows on the basis of quasi-hydrodynamic equations. Analysis of the calculation results suggests that, in the range of real microaccelerations, the 3D model does not add new effects to the structure of convective motion compared with simplified two-dimensional models. The conclusion is that simplified models can be beneficially used for the interpretation of measurements carried out with the DAKON convection sensor.     

Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm

A biofilm is usually defined as a layer of bacterial cells anchored to a surface. These cells are embedded into a polymer matrix that keeps them attached to each other and to a solid surface. Among a large variety of biofilms, in this paper we consider batch cultures. The mathematical model is formulated in terms of a quasilinear system of diffusion–reaction equations for biomass and nutrients concentrations, which exhibits possible degeneracy and singularities in the nonlinear diffusion coefficient. In the present paper, we propose a set of efficient numerical methods that speeds up the solution of the model. Mainly, Crank–Nicolson finite differences techniques for discretisation are combined with a Newton algorithm for the nonlinearities. Moreover, some numerical examples show the expected behaviour of the biomass and nutrients concentrations and also clearly illustrate some theoretically proved qualitative properties related to exponential decays or convergence to a critical biomass concentration depending on the values of the model parameters.     

Singular limit of a Navier–Stokes system leading to a free/congested zones two-phase model

The aim of this work is to justify mathematically the derivation of a viscous free/congested zones two-phase model from the isentropic compressible Navier–Stokes equations with a singular pressure playing the role of a barrier.     

Numerical bifurcation control of Mackey–Glass system

In this paper, a mathematical physiological model, Mackey–Glass system of a delay differential equation, is considered. With a greater delay, a periodic solution arises, which characterizes the disease of chronic granulocytic leukemia (CGL). To treat such disease, a blood transfusion feedback control is considered, from the point of view of mathematical control. By using a nonstandard finite-difference (NSFD) scheme to the control system, we obtain a numerical discrete system and analyze its Neimark–Sacker and fold bifurcations. The results imply that the condition of the illness could be relieved by transfusing blood to the patient, if the control is a delay control. Finally, the effectiveness of the control are illustrated by several numerical simulations.     

Multichannel type-I intermittency in two models of Rayleigh–Bénard convection

We report on the first observation of transitions to deterministic chaos via type-I intermittency with two channels of re-injection in two equivariant autonomous dynamical systems. First, we consider the standard Lorenz system which is equivariant under the action of the rotation of π around the z-axis. We also consider the same phenomenon in a nine-dimensional model of the Rayleigh–Bénard convection which is equivariant under the action of the Klein four group, of all isometries mapping a rectangle, which is not a square, on itself.     

Boundary Layer Heat Transport in turbulent Rayleigh-Bénard Convection in Air

We study the convective heat transfer in a large-scale Rayleigh-Bénard (RB) experiment which is called the “Barrel of Ilmenau”. We present the results of flow visualization and Particle Image Velocimetry (PIV) measurements of the near wall flow field in a plane perpendicular to the surface of the heated bottom plate. The experiment was run in a smaller rectangular inset that was placed inside the larger barrel. The Rayleigh number amounts to Ra = 1.4 × 10¹⁰. The aspect ratios were Γ_x = 1 in flow direction and Γ_y = 0.26 perpendicular to the vertical flow plane. The measurements have been undertaken using a 2 W continuous wave Laser in combination with a light sheet optics and various cameras. Due to the slender geometry of the cell, the mean wind is confined in one direction where the Laser light sheet is aligned parallel. The flow was seeded with droplets of 1...2 µm size generated using an ordinary fog machine. Flow visualization as well as the PIV data clearly show the intermittent character of the boundary layer flow field that permanently switches between “laminar” and “turbulent” phases. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)