Analytical Prediction of the Transition to Chaos in Porous Media Natural Convection

2008 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Porous Media ◽  
Natural Convection ◽  
Stability Analysis ◽  
Analytical Solution ◽  
Linear Stability Analysis ◽  
Transition Point ◽  
Analytical Prediction ◽  
Linear Solution ◽  
Transition To Chaos ◽  
Validity Condition

The failure of the linear stability analysis to predict accurately the transition point from steady to chaotic solutions in porous media natural convection motivates this study. A weak non-linear solution to the problem is shown to produce an accurate analytical expression for the transition point as long as the validity condition and consequent accuracy of the latter solution is fulfilled. The analytical results are compared to accurate computational solutions showing an excellent fit within the validity domain of the analytical solution.

Analytical Prediction of the Transition to Chaos in Lorenz System

2008 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Stability Analysis ◽  
Analytical Solution ◽  
Linear Stability Analysis ◽  
Transition Point ◽  
Lorenz System ◽  
Analytical Prediction ◽  
Lorenz Equations ◽  
Linear Solution ◽  
Transition To Chaos ◽  
Validity Condition

The failure of the linear stability analysis to predict accurately the transition point from steady to chaotic solutions in Lorenz equations motivates this study. A weak non-linear solution to the problem is shown to produce an accurate analytical expression for the transition point as long as the validity condition and consequent accuracy of the latter solution is fulfilled. The analytical results are compared to accurate computational solutions showing an excellent fit within the validity domain of the analytical solution.

scholarly journals Instability and Convection in Rotating Porous Media: A Review

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 147 ◽  
Author(s):  
Vadasz
Keyword(s):  
Porous Media ◽  
Natural Convection ◽  
Stability Analysis ◽  
Gravity Field ◽  
Linear Stability Analysis ◽  
Density Gradients ◽  
Centripetal Acceleration ◽  
Geostrophic Flows ◽  
Nonlinear Solutions ◽  
Rotating Porous Media

A review on instability and consequent natural convection in rotating porous media is presented. Taylor-Proudman columns and geostrophic flows exist in rotating porous media just the same as in pure fluids. The latter leads to a tendency towards two-dimensionality. Natural convection resulting from density gradients in a gravity field as well as natural convection induced by density gradients due to the centripetal acceleration are being considered. The former is the result of gravity-induced buoyancy, the latter is due to centripetally-induced buoyancy. The effect of Coriolis acceleration is also discussed. Linear stability analysis as well as weak nonlinear solutions are being derived and presented.

Weak Turbulence in Small Prandtl Number Convection in Porous Media

2010 ◽  
Author(s):  
Peter Vadasz
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Prandtl Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Weak Turbulence ◽  
Numerical Computations ◽  
Prandtl Numbers ◽  
Convection In Porous Media

The dynamics of weak turbulence in small Prandtl number convection in porous media is substantially distinct than the corresponding dynamics for moderate and large Prandtl numbers. Linear stability analysis is performed and its results compared with numerical computations to reveal the underlying phenomena.

Linear Stability Analysis

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Stability Analysis ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stability Problem ◽  
Linear Codes ◽  
Linear Equations ◽  
Critical Rayleigh Number ◽  
Linear Solution ◽  
Linear Stability Problem

This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.

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