Nonlinear Convection Equations Research Articles

Transition of thermally induced flow in a square cavity using an iterative predictor–corrector meshfree approach

AbstractThis paper aims to propose a new approach based on meshfree method to solve the dimensionless steady nonlinear natural convection equations. The solution methodology uses the moving least squares (MLS) shape functions to discretize the weak form of the governing equations and the Newton–Raphson algorithm to compute the nonlinear solution. Parametric studies of natural convection concerning closed two‐dimensional (2D) cavity with different operating conditions filled with air is used to assess the impact of the proposed approach on the performance of computed solutions. The cavity is formed by adiabatic and isothermal walls. For that, in all considered cases, the vertical sidewalls of the enclosure are heated differentially and maintained at hot and cold temperatures. The horizontal top and bottom walls are considered thermally insulated. The thermophysical properties of the working fluid are assumed to be constant except the density variations causing a body force term in the momentum equation. No‐slip boundary condition is applied at all enclosure boundaries and all internal body boundaries. The considered cases are described and analyzed in detail to check the validity of the proposed method. A comparison is performed between the results obtained from the present computer code and those available in the literature. The influence of various controlling parameters is analyzed and discussed graphically through the obtained streamlines, isotherms, and (maximum and average) Nusselt number. The numerical results computed by the proposed method are found to be in good agreement with the reference data.

Traveling wavefronts for density‐dependent diffusion reaction convection equation with time delay

We are concerned with the existence of traveling waves for density‐dependent diffusion reaction nonlinear convection equations with small time delay. We first study the existence and uniqueness of smooth and sharp‐type traveling wavefront solution of the wave speed c ≥ c∗ for equation without time delay, where c∗ is the minimal wave speed. Meanwhile, we construct a variational principle for c∗ from which the upper bound of c∗ is determined, while the lower bound of c∗ is attained by using the phase plane analysis. Then, we obtain the existence of smooth traveling wavefront solution for equation with small time delay for c > c∗ by applying the perturbation method and implicit function theorem.

Qualitative Analysis and Traveling wave Solutions for the Nonlinear Convection Equations with Absorption

We discuss qualitative behavior of the solutions for the nonlinear parabolic equation which modeling nonlinear convection equation with absorption. This model represents the movement of growing population that is ruled by convection process. In this paper, we concentrate on proving the existence of traveling wave solutions for the nonlinear convection-reaction equations. In addition, we consider the model when the speed of advective wave may breakdown and the problem has a shock wave solution. The mathematical interesting of the waves comes from the behaviors of singular differential equation and discussing the stability of the solution.

An Extended Theoretical Scenario for Classical Cepheids. I. Modeling Galactic Cepheids in the Gaia Photometric System

We present a new extended and detailed set of models for Classical Cepheid pulsators at solar chemical composition ($Z=0.02$, $Y=0.28$) based on a well tested nonlinear hydrodynamical approach. In order to model the possible dependence on crucial assumptions such as the Mass-Luminosity relation of central Helium burning intermediate-mass stars or the efficiency of superadiabatic convection, the model set was computed by varying not only the pulsation mode and the stellar mass but also the Mass-Luminosity relation and the mixing length parameter that is used to close the system of nonlinear hydrodynamical and convective equations. The dependence of the predicted boundaries of the instability strip as well as of both light and radial velocity curves on the assumed Mass-Luminosity and the efficiency of superadiabatic convection is discussed. Nonlinear Period-Mass-Luminosity-Temperature, Period-Radius and Period-Mass-Radius relations are also computed. The theoretical atlas of bolometric light curves for both the fundamental and first overtone mode has been converted in the Gaia filters $G$, $G_{BP}$ and $G_{BR}$ and the corresponding mean magnitudes have been derived. Finally the first theoretical Period-Luminosity-Color and Period-Wesenheit relations in the Gaia filters are provided and the resulting theoretical parallaxes are compared with Gaia Data Release 2 results for both fundamental and first overtone Galactic Cepheids.

Solutions to Painleve III and other nonlinear equations by a generalized Cole–Hopf transformation

In this paper, we introduce a generalized form of Cole‐Hopf transformation and apply it to find new closed‐form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.

Avalanche photon cooling by induced Compton scattering: Higher-order Kompaneets equation

Induced Compton scattering (ICS) is an interaction between intense electro-magnetic radiations and plasmas, where ICS transfers the energy from photons to plasmas. Although ICS is important for laser plasma interactions in laboratory experiments and for radio emission from pulsars propagating in pulsar wind plasmas, the detail of photon cooling process has not been understood. The problem is that, when ICS dominates, evolution of photon spectra is described as a nonlinear convection equation, which makes photon spectra to be multi-valued. Here, we propose a new approach to treat evolution of photon spectra affected by ICS. Starting from the higher-order Kompaneets equation, we find a new equation that resolves the unphysical behavior of photon spectra. In addition, we find the steady-state analytic solution, which is linearly stable. We also successfully simulate the evolution of photon spectra without artificial viscosity. We find that photons rapidly lose their energy by ICS with continuously forming solitary structures in frequency-space. The solitary structures have the logarithmically same width characterized by an electron temperature. The energy transfer from photons to plasma is more effective for broader spectrum of photons such as expected in astrophysical situations.

A Fast Explicit Operator Splitting Method for Modified Buckley–Leverett Equations

In this paper, we propose a fast explicit operator splitting method to solve the modified Buckley---Leverett equations which include a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. The method splits the original equation into two equations, one with a nonlinear convective term and the other one with high-order linear terms so that appropriate numerical methods can be applied to each of the split equations: the high-order linear equation is numerically solved using a pseudo-spectral method, while the nonlinear convective equation is integrated using the Godunov-type central-upwind scheme. The spatial order of the central-upwind scheme depends on the order of the piecewise polynomial reconstruction: we test both the second-order minmod-based reconstruction and fifth-order WENO5 one to demonstrate that using higher-order spatial reconstruction leads to more accurate approximation of solutions. A variety of numerical examples in both one and two space dimensions show that the solutions may have many different saturation profiles depending on the initial conditions, diffusion parameter, and the third-order mixed derivatives parameter. The results are consistent with the study of traveling wave solutions and their bifurcation diagrams.

Nonisothermal nematic liquid crystal flows with the Ball–Majumdar free energy

In this paper, we prove the existence of global-in-time weak solutions for an evolutionary PDE system modelling nonisothermal Landau–de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier–Stokes system for the macroscopic velocity $$\mathbf{u}$$ is coupled to a nonlinear convective parabolic equation describing the evolution of the $$Q$$ -tensor $$\mathbb {Q}$$ , namely a tensor-valued variable representing the normalized second-order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $$\vartheta $$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $$f$$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the $$Q$$ -tensor equation, a term that is at the same time singular in $$\mathbb {Q}$$ and degenerate in $$\vartheta $$ . To treat it, a careful analysis of the properties of $$f$$ , particularly of its blow-up rate, is carried out.

A transport-physics-accurate convection discretization based on first order upwind and the need for a new error assessment strategy

Several numerical schemes have been proposed in the literature to deal with convection phenomena, but all of them present some spurious effects. In this paper we show that, once a more physically-oriented error assessment tool is used in substitution for the classical wave number analysis, it is possible to develop an improved family of numerical schemes. Results are presented for a particular implementation based on first order upwind, in test cases involving linear and non-linear convective equation. This new scheme is able to simulate linear convection equation under uniform field with machine accuracy precision, with zero diffusion and dispersion. Moreover, the results for the non-linear convection are also significantly more accurate over the whole spectrum when compared to a traditional scheme based on a fifteen points stencil.

Comparison of different order Adams-Bashforth methods in an atmospheric general circulation model

The Asselin-Robert time filter used in the leapfrog scheme does degrade the accuracy of calculations. As an attractive alternative to leapfrog time differencing, the second-order Adams-Bashforth method is not subject to time splitting instability and keeps excellent calculation accuracy. A second-order Adams-Bashforth model has been developed, which represents better stability, excellent convergence and improved simulation of prognostic variables. Based on these results, the higher-order Adams-Bashforth methods are developed on the basis of NCAR (National Center for Atmospheric Research) CAM 3.1 (Community Atmosphere Model 3.1) and the characteristics of dynamical cores are analyzed in this paper. By using Lorenz nonlinear convective equations, the filtered leapfrog scheme shows an excellent pattern for eliminating 2Δt wave solutions after 20 steps but represents less computational solution accuracy. The fourth-order Adams-Bashforth method is closely converged to the exact solution and provides a reference against which other methods may be compared. Thus, the Adams-Bashforth methods produce more accurate and convergent solution with differencing order increasing. The Held-Suarez idealized test is carried out to demonstrate that all methods have similar climate states to the results of many other global models for long-term integration. Besides, higher-order methods perform better in mass conservation and exhibit improvement in simulating tropospheric westerly jets, which is likely equivalent to the advantages of increasing horizontal resolutions. Based on the idealized baroclinic wave test, a better capability of the higher-order method in maintaining simulation stability is convinced. Furthermore, after the baroclinic wave is triggered through overlaying the steady-state initial conditions with the zonal perturbation, the higher-order method has a better ability in the simulation of baroclinic wave perturbation.

Explicit traveling wave solutions to nonlinear evolution equations

First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.

An oscillation-free high order TVD/CBC-based upwind scheme for convection discretization

An oscillation-free high order scheme is presented for convection discretization by using the normalized-variable formulation in the finite volume framework. It adopts the cubic upwind interpolation scheme as the basic scheme so as to obtain high order accuracy in smooth solution domain. In order to avoid unphysical oscillations of numerical solutions, the present scheme is designed on the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results of several linear and nonlinear convection equations with smooth or discontinuous initial distributions demonstrate the present scheme possesses second-order accuracy, good robustness and high resolution.

Time-reversibility of the Euler equations as a benchmark for energy conserving schemes

The three-dimensional incompressible Euler equations are time-reversible. This property should be preserved as well as possible by numerical discretizations. This article investigates the time-reversibility properties of various solvers designed for incompressible Navier–Stokes computations. The test case is the inviscid Taylor–Green vortex, which becomes “turbulent” before the time is reversed to try to recover the initial condition. The simulations are performed using high and low order finite difference solvers as well as using a pseudo-spectral solver. Various time-stepping schemes are also investigated. Although the flow statistics are significantly affected by the accuracy of the space discretization, the time-reversibility is not because most space-discretizations are time-reversible for an exact time-stepping. The crucial factor for time-reversibility is the accuracy of the time-stepping scheme and its interaction with the space-discretization. Furthermore, an important practical requirement for the solver is to be energy conserving in order to avoid numerical instability. An energy conserving solver using an accurate time-stepping is then able to go back almost perfectly from a complex “turbulent” flow to the simple initial condition. Therefore, we propose that this constitutes a severe and useful benchmark that Navier–Stokes solvers should challenge. The present investigations and their conclusions are also supported by parallel 1-D investigations, using the non-linear convection equation (inviscid Burgers) and the linear convection equation.

Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations

Recently, Rajabi et al. (Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570–573.) discussed the solutions of temperature distribution in lumped system of combined convection–radiation. They solved a nonlinear equation of the steady conduction in a slab with variable thermal conductivity using both perturbation and homotopy perturbation methods. They claim that homotopy perturbation method (HPM) does not require any small parameter. However, this statement is not true always. Moreover, HPM have no criteria for establishing the convergence of the series solution. In this letter we have explicitly shown that the results of the problem considered in example 2 of (Rajabi, Ganji, Therian, Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A 360 (2007) 570–573.) are valid only for 0 ⩽ ε ⩽ 1 . We have used the homotopy analysis method for finding the more meaningful solution.

Application of homotopy perturbation method in nonlinear heat conduction and convection equations

Abstract One of the newest analytical methods to solve nonlinear equations is the application of both homotopy and perturbation techniques. In this Letter, homotopy perturbation method (HPM), which does not need small parameters, is compared with the perturbation method in the field of heat transfer. The justification for using this method is the difficulties and limitations of perturbation and homotopy when used individually. In this research, homotopy perturbation method is used to solve for the temperature distribution in lumped system of combined convection–radiation and also a nonlinear equation of the steady conduction in a slab with variable thermal conductivity.

On unconditional conservation of kinetic energy by finite-difference discretizations of the linear and non-linear convection equation

In the past the development of kinetic energy conserving finite-difference methods mostly focused on second-order accurate central methods defined on uniform grids. Nowadays the need for high-order accurate discretizations, to perform for instance accurate numerical simulations of turbulent flow, calls for the development of novel kinetic energy conserving discretization schemes. Instead of choosing a fixed basis discretization up front, in this paper a different, more general, approach is applied. For a Cartesian mesh, sets of conditions are presented such that all discretizations of the linear or non-linear convection equation which obey these conditions, unconditionally conserve kinetic energy. For the linear convection equation it is shown that on a uniform grid it is necessary and sufficient for a discretization to be central in order to be fully conservative, that is: such discretizations not only unconditionally conserve kinetic energy but also unconditionally conserve momentum. On non-uniform grids an algorithm is introduced that can be used to generate fully conservative discretizations that are at least first-order accurate. The derivation of the discretization conditions for the non-linear convection equation is performed in the two-dimensional (2D) linear case. Some examples on uniform grids and on non-uniform grids are presented. It is shown that on uniform grids no upper limit exists with respect to the accuracy of the kinetic energy conserving method. For the higher-dimensional linear and non-linear convection equation the same set of conditions, which ensure the unconditional conservation of kinetic energy, are found as in the 2D linear case. Other results too are found to be straightforward generalizations of the corresponding 2D linear results. It is shown that the fourth-order unconditionally kinetic energy conserving discretization on a staggered mesh introduced in this paper is well suited to simulate the initial development of an inviscid shear layer instability in a divergence-free flow.

Analytic solutions of a nonlinear convective equation in population dynamics

Analytic solutions are presented for a simple nonlinear convective equation of use in population dynamics. In spite of its simplicity the equation predicts rich behavior including a velocity inversion transition. Stability considerations are also presented.

Nonlinear Convection in a Rotating Annulus with a Finite Gap

Thermal convection in a fluid-filled gap between the two corotating, concentric cylindrical sidewalls with sloping curved ends driven by radial buoyancy was first studied by Busse (Busse, F.H., "Thermal instabilities in rapidly rotating systems", J. Fluid Mech . 44 , 441-460 (1970)). The annulus model captures the key features of rotating convection in full spherical geometry and has been widely employed to study convection, magnetoconvection and dynamos in planetary systems, usually in connection with the small-gap approximation neglecting the effect of azimuthal curvature of the annulus. This article investigates nonlinear thermal convection in a rotating annulus with a finite gap through numerical simulations of the full set of nonlinear convection equations. Three representative cases are investigated in detail: a large-gap annulus with the ratio of the radii ( s i and s o ) of the sidewalls ξ = s i / o s = 0.1, a medium-gap annulus with ξ = 0.35 and a small-gap annulus with ξ = 0.8. Near the onset of convection, the effect of rapid rotation through the sloping ends forces the first (Hopf) bifurcation in the form of small-scale, steadily drifting rolls (thermal Rossby waves). At moderately large Rayleigh numbers, a variety of different convection patterns are found, including mixed-mode steadily drifting, quasi-periodic (vacillating) and temporally chaotic convection in association with various temporal and spatial symmetry-breaking bifurcations. Our extensive simulations suggest that competition between nonlinear and rotational effects with increasing Rayleigh number leads to an unusual sequence of bifurcation characterized by enlarging the spatial scale of convection.

Null controllability of nonlinear convective heat equations

The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in Lk .

The effect of a local increase in impurity concentration in one-dimensional hydrodynamics

The system of non-linear equations of one-dimensional convective diffusion of a passive impurity in the case of a flow of fluid described by the model Burgers equation is solved using Cole-Hopf and Darboux transformations. For equal kinetic coefficients of the fluid its solution is reduced to solving linear heat-conduction equations. For integer Prandtl numbers, differing from unity, this reduction is successful for flows of uniformly moving shock waves, smoothed by the viscosity effect. Its possibility is closely related to the factorizability of second-order differential operators in this case, their decomposability into the product of first-order operators, and the additional internal symmetry (supersymmetry) of the problem. The interaction of shock waves and the impurity solitons they transfer has an absolutely inelastic character. A local increase in the impurity concentration is a result of the merging of impurity solitons. It is pointed out that, for certain equations of state, the reduction of the solutions of the non-linear convective diffusion equations to the solution of linear heat-conduction equations is also possible for an active impurity (when the kinetic coefficients of the fluid are equal).