scholarly journals A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-Bénard Convection

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jitender Singh
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Shooting Method ◽  
Fluid Layer ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type is considered where components of are known at one of the boundaries and components of are specified at the other boundary. The map is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of Ha (2001) to the BVPs arising in the higher order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-Bénard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role.

Rayleigh-Bénard Convection in a Vertically Oscillated Fluid Layer

2000 ◽  
Vol 84 (1) ◽  
pp. 87-90 ◽  
Author(s):  
Jeffrey L. Rogers ◽  
Michael F. Schatz ◽  
Jonathan L. Bougie ◽  
Jack B. Swift
Keyword(s):  
Fluid Layer ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

scholarly journals Bounds on Rayleigh–Bénard convection with imperfectly conducting plates

2010 ◽  
Vol 665 ◽  
pp. 158-198 ◽  
Author(s):  
RALF W. WITTENBERG
Keyword(s):  
Rayleigh Number ◽  
Biot Number ◽  
Fluid Layer ◽  
Temperature Drop ◽  
Upper Bounds ◽  
Fixed Temperature ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh–Bénard convection on analytical upper bounds on convective heat transport. We model imperfectly conducting bounding plates in two ways: using idealized mixed thermal boundary conditions (BCs) of constant Biot number η, continuously interpolating between the previously studied fixed temperature (η = 0) and fixed flux (η = ∞) cases; and by explicitly coupling the evolution equations in the fluid in the Boussinesq approximation through temperature and flux continuity to identical upper and lower conducting plates. In both cases, we systematically formulate a bounding principle and obtain explicit upper bounds on the Nusselt numberNuin terms of the usual Rayleigh numberRameasuring the average temperature drop across the fluid layer, using the ‘background method’ developed by Doering and Constantin. In the presence of plates, we find that the bounds depend on σ =d/λ, wheredis the ratio of plate to fluid thickness and λ is the conductivity ratio, and that the bounding problem may be mapped onto that for Biot number η = σ. In particular, for each σ > 0, for sufficiently largeRa(depending on σ) we show thatNu≤c(σ)R1/3≤CRa1/2, whereCis a σ-independent constant, and where the control parameterRis a Rayleigh number defined in terms of the full temperature drop across the entire plate–fluid–plate system. In theRa→ ∞ limit, the usual fixed temperature assumption is a singular limit of the general bounding problem, while fixed flux conditions appear to be most relevant to the asymptoticNu–Rascaling even for highly conducting plates.

scholarly journals TEMPERATURE MODULATION IN RAYLEIGH–BÉNARD CONVECTION

2008 ◽  
Vol 50 (2) ◽  
pp. 231-245 ◽  
Author(s):  
JITENDER SINGH ◽  
RENU BAJAJ
Keyword(s):  
Floquet Theory ◽  
Fluid Layer ◽  
Temperature Modulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Horizontal Fluid Layer ◽  
Time Periodic

AbstractThe stability characteristics of an infinite horizontal fluid layer excited by a time-periodic, sinusoidally varying free-boundary temperature, have been investigated numerically using the Floquet theory. It has been found that the modulation of the temperature gradient across the fluid layer affects the onset of the Rayleigh–Bénard convection. Modulation can give rise to instability in the subcritical conditions and it can also suppress the instability in the supercritical cases. The instability in the fluid layer manifests itself in the form of either a harmonic or subharmonic flow, controlled by thermal modulation.

Effects of Coriolis force and nonuniform temperature gradient on the Rayleigh–Benard convection

1986 ◽  
Vol 64 (1) ◽  
pp. 90-99 ◽  
Author(s):  
N. Rudraiah ◽  
O. P. Chandna
Keyword(s):  
Temperature Gradient ◽  
Coriolis Force ◽  
Fluid Layer ◽  
Nonuniform Temperature ◽  
Bénard Convection ◽  
Benard Convection ◽  
Bounding Surfaces ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Good Agreement

The effects of the Coriolis force and a nonuniform temperature gradient on the onset of the Rayleigh–Benard convection in a thin, horizontal, rotating fluid layer is studied using linear-stability analysis. It is shown analytically that the method and rate of heating, the Coriolis force, and the nature of the bounding surfaces of the fluid layer significantly influence the value of the Rayleigh number at the onset of marginal convection. The mechanism for suppressing or augmenting convection is discussed in detail. The Galerkin technique employed here is much easier to use than that the method of Chandrasekhar (5). The analytical results obtained from using this procedure are compared with the published experimental data and the results obtained from numerical procedures; good agreement is found.

Roll-pattern evolution in finite-amplitude Rayleigh–Beńard convection in a two-dimensional fluid layer bounded by distant sidewalls

1984 ◽  
Vol 143 ◽  
pp. 125-152 ◽  
Author(s):  
P. G. Daniels
Keyword(s):  
Rayleigh Number ◽  
Temporal Evolution ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).

scholarly journals Steady Rayleigh–Bénard convection between no-slip boundaries

2021 ◽  
Vol 933 ◽  
Author(s):  
Baole Wen ◽  
David Goluskin ◽  
Charles R. Doering
Keyword(s):  
Heat Transport ◽  
Fluid Layer ◽  
Turbulent Convection ◽  
Bénard Convection ◽  
Benard Convection ◽  
Convection Rolls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Steady Convection ◽  
Asymptotic Scaling

The central open question about Rayleigh–Bénard convection – buoyancy-driven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling.

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