Nonlinear Convection Problems Research Articles

Modeling of heat transfer at local convection over a vertical throughflow in a two-layered air-heat generating porous system

Convective heat transfer in an air layer partially filled with a heat-generating granular porous medium is studied. There is a slow seepage of air through the layer in the vertical direction with a constant velocity. Equal temperatures are maintained at the outer solid permeable boundaries, and the heat source strength is constant within the porous sublayer and is proportional to the solid volume fraction. The permanent heat generation within the porous sublayer, combined with the vertical throughflow, causes a nonlinear thermal profile which is conducive for convection to occur. The Boussinesq approximation and Darcy's law are used to describe this convection. Numerical solution of the nonlinear convective problem is obtained on the basis of Newton's method. In the limiting case, the numerical data for the onset of convection are compared with the results of the earlier paper of the authors, where a linear stability theory and a method for constructing of the fundamental system of partial solution vectors have been applied, and with the data by other authors. The stationary regimes of local convection, which occurs in an “air – heat-generating porous medium-air” system over the basic vertical throughflow, and its effect on the heat transfer from the porous air sublayer with the growth of supercriticality are studied. It is shown that, depending on the velocity of the basic throughflow (the Peclet number), convection excitation can be both soft (due to supercritical pitchfork bifurcation) and hard (when the loss of stability of the basic throughflow is accompanied by subcritical pitchfork bifurcation that gives rise to an unstable secondary convective regime). This secondary regime is replaced by a stable tertiary convective regime with increasing supercriticality. It has been found that the total heat transfer rate for the upward basic throughflow exceeds that for the downward basic throughflow significantly, and that local convection at any direction of this throughflow increases the heat transfer rate in the system. An increase in the Nusselt number with the growth of supercriticality is recorded. However, a noticeable contribution of local convection to the total heat transfer is observed only when all values of the Pecklet number are negative and its positive values are lower than 2.

Finite element error estimates for nonlinear convective problems

AbstractThis paper is concerned with the analysis of the finite element method applied to a nonstationary nonlinear convective problem. Using special estimates of the convective terms, we prove

Curvilinear finite-volume schemes using high-order compact interpolation

During the last years, the need of high fidelity simulations on complex geometries for aeroacoustics predictions has grown. Most of high fidelity numerical schemes, in terms of low dissipative and low dispersive effects, lie on finite-difference (FD) approach. But for industrial applications, FD schemes are less robust compared to finite-volume (FV) ones. Thus the present study focuses on the development of a new compact FV scheme for two- and three-dimensional applications. The proposed schemes are formulated in the physical space and not in the computational space as it is the case in most of the known works. Therefore, they are more appropriate for general grids. They are based on compact interpolation to approximate interface-averaged field values using known cell-averaged values. For each interface, the interpolation coefficients are determined by matching Taylor series expansions around the interface center. Two types of schemes can be distinguished. The first one uses only the curvilinear abscissa along a mesh line to derive a sixth-order compact interpolation formulae while the second, more general, uses coordinates in a spatial three-dimensional frame well chosen. This latter is formally sixth-order accurate in a preferred direction almost orthogonal to the interface and at most fourth-order accurate in transversal directions. For non-linear problems, different approaches can be used to keep the high-order scheme. However, in the present paper, a MUSCL-like formulation was sufficient to address the presented test cases. All schemes have been modified to treat multiblock and periodic interfaces in such a way that high-order accuracy, stability, good spectral resolution, conservativeness and low computational costs are guaranteed. This is a first step to insure good scalability of the schemes although parallel performances issues are not addressed. As high frequency waves, badly resolved, could be amplified and then destabilize the scheme, compact filtering operators have been used. Numerous test cases as the linear convection of a Gaussian wave, the convection of a Lamb–Oseen vortex and the diffraction of an acoustic wave on a plane have been realized to validate the schemes. The most efficient schemes are shown to be at least fifth-order accurate on linear and non-linear convection problems. They are also less dissipative and less dispersive on non-uniform curvilinear grids than schemes using implicit interpolation with constant coefficients of the same order on uniform cartesian grids.

Stable and Accurate Artificial Dissipation

Stability for nonlinear convection problems using centered difference schemes require the addition of artificial dissipation. In this paper we present dissipation operators that preserve both stability and accuracy for high order finite difference approximations of initial boundary value problems.

On nonlinear convection in mushy layers Part 1. Oscillatory modes of convection

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

A Level-Set Method for Computing Solutions to Viscoelastic Two-Phase Flow

A finite-element code based on the level-set method is developed for simulating the motion of viscoelastic two-phase flow problems. This method is a generalization of the finite-difference approach described in [1–4] for computing solutions to two-phase problems of inviscid and viscous fluids. The Marchuk–Yanenko operator-splitting technique is used to decouple the difficulties associated with the nonlinear convection term, the incompressibility constraint, the viscoelastic term, and the interface motion problem. The nonlinear convection problem is solved using a least-squares conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The code is verified by performing a convergence study to show that the results are independent of the mesh and time-step sizes. Using our code we have studied the deformation of drops in simple shear and pressure-driven flows and of bubbles in gravity-driven flows over a wide range of dimensionless capillary (Ca) and Deborah numbers (De). For a Newtonian bubble rising in a quiescent viscoelastic liquid we find that there are limiting values of the parameters De and Ca, above which the bubble assumes a characteristic shape with a cusp-like trailing edge. The front of the bubble, however, remains round, as the local viscoelastic and viscous stresses act to round the bubble. In a pressure-driven flow the drop is stretched so that its front, which is closer to the channel center, remains round, and the trailing edge, which is closer to the channel wall, becomes sharp. These numerical results are in agreement with the experimental observations.

Critical thresholds in Euler-Poisson equations

We present a preliminary study of a new phenom- ena associated with the Euler-Poisson equations — the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic,ON1O critical thresh- old. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problems with or without various forcing mech- anisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical thresh- old which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock dis- continuity, except for the non-generic case of increasing initial profile,u 0 0. In contrast, we show that the corresponding one- dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initialN0;u 0O-configuration satisfiesu 0 p 2k0 - see (2.11) below, allowing a finite, crit- ical negative velocity gradient. As is typical for such nonlinear convection problems, one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenom- ena.

A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows

A distributed Lagrange multiplier/fictitious domain method (DLM) is developed for simulating the motion of rigid particles suspended in the Oldroyd-B fluid. This method is a generalization of the one described in [R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, Int. J. Multiphase Flows 25 (1998) 755–794] where the motion of particles suspended in a Newtonian fluid was simulated. In our implementation of the DLM method, the fluid–particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing equations for the Oldroyd-B liquid are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. We use the Marchuk–Yanenko operator-splitting technique to decouple the difficulties associated with the incompressibility constraint, nonlinear convection and viscoelastic terms. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The nonlinear convection problem is solved using a least square conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The code is verified performing a convergence study to show that the results are independent of the mesh and time step sizes. Our simulations show that, when particles are dropped in a channel, and the viscoelastic Mach number ( M) is less than 1 and the elasticity number ( E) is greater than 1, the particles chain along the flow direction; this agrees with the results presented in [P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–325]. In our simulations of the fluidization of 102 particles in a two-dimensional bed, we find that the particles near the channel walls form chains that are parallel to the walls, but the distribution of particles away from the walls is relatively random.

Optimal order spline methods for nonlinear differential and integro-differential equations

In this work, we analyze new robust spline approximation methods for mth order boundary value problems described by nonlinear ordinary differential and integro-differential equations with m linear boundary conditions. Our main aim is to introduce a cost-effective alternative to the highly successful orthogonal collocation method, and to prove stability and convergence properties similar to the orthogonal collocation method. Our method is a discrete Petrov-Galerkin method, in which we seek a spline approximation u h of order m + r, but with one higher degree of smoothness than for the standard orthogonal collocation approximation: we require u h ∈ C m (in contrast to C m − 1 in the orthogonal collocation case), so that u h ( m) ∈ C. We show that optimal order convergence and superconvergence at break points hold without any mesh restriction, provided only that a certain underlying quadrature rule has degree of precision at least 2 r − 1. There is one respect in which the present method has an advantage over orthogonal collocation. Firstly, because of the increased continuity of u h , the number of unknown variables is reduced (almost halved in certain practical cases), which is a significant reduction for complex nonlinear problems. Orthogonal collocation method itself can be obtained by changing a few parameters in our method. To test our theoretical results and demonstrate the generality of the method, we consider an application to a catalytic combustion model problem involving nonlinear integro-differential equation, compute solutions of a nonlinear ordinary differential equation, and solve a boundary value problem with a thin boundary layer occurring in a nonlinear convection problem.

Nonlinear convection with variable coefficient of thermal expansion

The problem of nonlinear convection in a horizontal layer of fluid with variable coefficient of thermal expansion is considered. It is found that square cells transport always more heat than two-dimensional rolls and that rolls are always unstable. Variable coefficient of thermal expansion can strongly affect the critical Rayleigh number and the horizontal scale of the motion. Subcritical instabilities can exist which are associated with either hexagon pattern convection or square pattern convection. Particular forms of the coefficient thermal expansion are essential to determine the preferred flow patterns which could be of the forms of squares or hexagons with either upward of downward motion at the cells' centers.

Nonlinear convection in the presence of a vertical magnetic field

The present investigation provides a comparison between the results based on analytical approximation method and numerical methods to the problem of nonlinear convection, in the presence of a vertical magnetic field. There is a fair agreement between the two methods and the analytical approximation method yieldsR 1/3 law for the Nusselt number. The results of the analytical method show that for a given value ofQ heat convection attains a constant value at large values ofα and finally becomes independent ofα.

A numerical study of non-linear convection in a compressible medium

In this paper we consider the problem of non-linear convection in a compressible layer with polytropic structure. After deriving the appropriate forms of the basic conservation equations a single mode expansion is used to obtain a simpler model, within the framework of the anelastic approximation. These equations are integrated, using a band-matrix algorithm, for two characteristic density stratifications. The results are given in the form of graphs and are discussed in detail. An outline of the numerical algorithm is given in an appendix together with a discussion of its effectiveness.

Explicit finite difference scheme for nonlinear convective problems

An explicit finite difference scheme is proposed for nonlinear convective problems. The procedure is applied to a nonlinear convective diffusion problem with a nonlinear boundary condition. It is shown that the nonlinear boundary condition may be treated directly without linearization which has to be carried out in the implicit method to avoid a nonlinear system of algebraic equations. The explicit procedure is more direct and less cumbersome than the implicit procedure and the answers are in excellent agreement with those of the implicit method.

Non-linear Convection with Free Boundaries

In an IAU Symposium report, Spiegel has made mention of the basic reasons for the astrophysical interest in the problem of non-linear convection of a horizontal fluid layer heated from below. Unfortunately the two important parameters of this problem—the Prandtl number P and the Rayleigh number R take on ‘extreme’ values in the astrophysical situation and this presents immediate difficulties as far as the numerical solution of the basic equations go.

A stability criterion for steady finite amplitude convection with an external magnetic field

In a horizontal layer of fluid, thermal expansion or the presence of dissolved salt may cause a density gradient opposite to the direction of gravity. In such cases, when the buoyancy forces are sufficient to overcome the dissipative effects, the static state becomes unstable and convective motions arise. If the layer is infinitely large in horizontal extent, the non-linear convection problem is highly degenerate, admitting many different steady-state solutions. A general necessary criterion for stability of such non-linear steady solutions is developed here for the case in which a homogeneous vertical magnetic field acts on the fluid. The criterion is demonstrated for two rigid bounding surfaces which are perfect thermal and electrical conductors, but it is applicable to more general kinds of boundary conditions.