Fundamentals of laminar free convection in internally heated fluids at values of the Rayleigh–Roberts number up to

2018 ◽  
Vol 846 ◽  
pp. 966-998 ◽  
Author(s):  
Kenny Vilella ◽  
Angela Limare ◽  
Claude Jaupart ◽  
Cinzia G. Farnetani ◽  
Loic Fourel ◽  
...  
Keyword(s):  
Scaling Laws ◽  
Flow Characteristics ◽  
Internal Heat ◽  
Three Dimensions ◽  
Time Dependent ◽  
Free Boundaries ◽  
Cylindrical Structures ◽  
Thermal Anomalies ◽  
Split And Merge ◽  
Isothermal Fluid

Motions in the solid mantle of silicate planets are predominantly driven by internal heat sources and occur in laminar regimes that have not been systematically investigated. Using high-resolution numerical simulations conducted in three dimensions for a large range of Rayleigh–Roberts numbers ($5\times 10^{3}\leqslant Ra_{H}\leqslant 10^{9}$), we have determined the characteristics of flow in internally heated fluid layers with both rigid and free slip boundaries. Superficial planforms evolve with increasing $Ra_{H}$ from a steady-state tessellation of hexagonal cells with axial downwellings to time-dependent clusters of thin linear downwellings within large areas of nearly isothermal fluid. The transition between the two types of planforms occurs as a remarkable flow polarity reversal over a small $Ra_{H}$ range, such that downwellings go from isolated cylindrical structures encircled by upwellings to thin interconnected linear segments outlining polygonal cells. In time-dependent regimes at large values of $Ra_{H}$, linear downwellings dominate the flow field at shallow depth but split and merge at intermediate depths into nearly cylindrical plume-like structures that go through the whole layer. With increasing $Ra_{H}$, the number of plumes per unit area and their velocities increase whilst the amplitude of thermal anomalies decreases. Scaling laws for the main flow characteristics are derived for $Ra_{H}$ values in a $10^{6}$–$10^{9}$ range. For given $Ra_{H}$, plumes are significantly colder, narrower and wider apart beneath free boundaries than beneath rigid ones. From the perspective of planetary studies, these results alert to the dramatic changes of convective planform that can occur along secular cooling.

BAYESIAN ESTIMATION OF A TIME DEPENDENT INTERNAL HEAT SOURCE IN A HETEROGENEOUS MEDIA

2016 ◽  
Author(s):  
Carolina Palma Naveira Cotta ◽  
Kelvin Chen ◽  
Christopher Tostado ◽  
Philippe Rollemberg d'Egmont ◽  
Fernando Duda ◽  
...  
Keyword(s):  
Heat Source ◽  
Bayesian Estimation ◽  
Heterogeneous Media ◽  
Internal Heat ◽  
Internal Heat Source ◽  
Time Dependent

Evidence for the homothetic scaling of urban forms

2018 ◽  
Vol 47 (5) ◽  
pp. 870-888 ◽  
Author(s):  
Rémi Lemoy ◽  
Geoffrey Caruso
Keyword(s):  
Land Use ◽  
Population Density ◽  
Scaling Laws ◽  
Radial Distance ◽  
Three Dimensions ◽  
City Size ◽  
Cube Root ◽  
Radial Profiles ◽  
Population Grid ◽  
Urban Scaling

The size and form of cities influence their social and environmental impacts. Whether cities have the same form irrespective of their size is still an open question. We analyse the profile of artificial land and population density, with respect to the distance to their main centre, for the 300 largest European cities. Our analysis combines the GMES/Copernicus Urban Atlas 2006 land use database at 5 m resolution for 300 larger urban zones with more than 100,000 inhabitants and the Geostat population grid at 1 km resolution. We find a remarkable constancy of radial profiles across city sizes. Artificial land profiles scale in the two horizontal dimensions with the square root of city population, while population density profiles scale in three dimensions with its cube root. In short, cities of different size are homothetic in terms of land use and population density, which challenges the idea that larger cities are more parsimonious in the use of land per capita. While earlier literature documented the scaling of average densities (total surface and population) with city size, we document the scaling of the whole radial distance profile with city size, thus liaising intra-urban radial analysis and systems of cities. Our findings also yield homogenous spatial definitions of cities, from which we can re-question urban scaling laws and Zipf’s law for cities.

Critical dynamics and renormalization group (RG)

2021 ◽  
pp. 875-898
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Renormalization Group ◽  
Time Evolution ◽  
Langevin Equation ◽  
Scaling Laws ◽  
Transport Coefficients ◽  
Time Dependent ◽  
Critical Systems ◽  
Static Properties ◽  
Critical Domain ◽  
Dynamical Scaling

Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.

Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates

2018 ◽  
Author(s):  
Lyudmyla Barannyk ◽  
John Crepeau ◽  
Patrick Paulus ◽  
Ali Siahpush
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Heat Generation ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Time Dependent ◽  
Cylindrical Coordinates ◽  
System Approaches ◽  
Melting And Solidification

A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model differs from the approximate model since it incorporates the time-dependent terms for small times, and as the system approaches steady-state, the curves converge. At higher Stefan numbers, the system approaches steady-state faster than for lower Stefan numbers. During the transient process for both melting and solidification, the temperature profiles become parabolic.

A bound on the vertical transport of heat in the ‘ultimate’ state of slippery convection at large Prandtl numbers

2013 ◽  
Vol 729 ◽  
pp. 103-122 ◽  
Author(s):  
Xiaoming Wang ◽  
Jared P. Whitehead
Keyword(s):  
Quadratic Form ◽  
Boundary Conditions ◽  
Heat Transport ◽  
Upper Bound ◽  
Vertical Transport ◽  
Three Dimensions ◽  
Free Boundaries ◽  
Ultimate State ◽  
Vertical Heat ◽  
Prandtl Numbers

AbstractAn upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’$R{a}^{1/ 2} $scaling and, in the limit of infinite$Pr$, agrees with the bound of$R{a}^{5/ 12} $recently derived in that limit for stress-free boundaries.

scholarly journals A Variational Formulation of Nonequilibrium Thermodynamics for Discrete Open Systems with Mass and Heat Transfer

Entropy ◽  
2018 ◽  
Vol 20 (3) ◽  
pp. 163 ◽  
Author(s):  
François Gay-Balmaz ◽  
Hiroaki Yoshimura
Keyword(s):  
Heat Transfer ◽  
Entropy Production ◽  
Variational Formulation ◽  
Evolution Equations ◽  
Nonequilibrium Thermodynamics ◽  
Open Systems ◽  
Internal Heat ◽  
Nonholonomic Constraints ◽  
Time Dependent ◽  
Nonlinear Constraint

We propose a variational formulation for the nonequilibrium thermodynamics of discrete open systems, i.e., discrete systems which can exchange mass and heat with the exterior. Our approach is based on a general variational formulation for systems with time-dependent nonlinear nonholonomic constraints and time-dependent Lagrangian. For discrete open systems, the time-dependent nonlinear constraint is associated with the rate of internal entropy production of the system. We show that this constraint on the solution curve systematically yields a constraint on the variations to be used in the action functional. The proposed variational formulation is intrinsic and provides the same structure for a wide class of discrete open systems. We illustrate our theory by presenting examples of open systems experiencing mechanical interactions, as well as internal diffusion, internal heat transfer, and their cross-effects. Our approach yields a systematic way to derive the complete evolution equations for the open systems, including the expression of the internal entropy production of the system, independently on its complexity. It might be especially useful for the study of the nonequilibrium thermodynamics of biophysical systems.

scholarly journals Convection in an internally heated stratified heterogeneous reservoir

2019 ◽  
Vol 870 ◽  
pp. 67-105 ◽  
Author(s):  
Angela Limare ◽  
Claude Jaupart ◽  
Edouard Kaminski ◽  
Loic Fourel ◽  
Cinzia G. Farnetani
Keyword(s):  
Heat Source ◽  
Scaling Laws ◽  
Lower Layer ◽  
Thermal Structure ◽  
Internal Heat ◽  
Source Distribution ◽  
Heat Sources ◽  
Homogeneous Fluid ◽  
Earth’S Mantle ◽  
Lower Fluid

The Earth’s mantle is chemically heterogeneous and probably includes primordial material that has not been affected by melting and attendant depletion of heat-producing radioactive elements. One consequence is that mantle internal heat sources are not distributed uniformly. Convection induces mixing, such that the flow pattern, the heat source distribution and the thermal structure are continuously evolving. These phenomena are studied in the laboratory using a novel microwave-based experimental set-up for convection in internally heated systems. We follow the development of convection and mixing in an initially stratified fluid made of two layers with different physical properties and heat source concentrations lying above an adiabatic base. For relevance to the Earth’s mantle, the upper layer is thicker and depleted in heat sources compared to the lower one. The thermal structure tends towards that of a homogeneous fluid with a well-defined time constant that scales with $Ra_{H}^{-1/4}$, where $Ra_{H}$ is the Rayleigh–Roberts number for the homogenized fluid. We identified two convection regimes. In the dome regime, large domes of lower fluid protrude into the upper layer and remain stable for long time intervals. In the stratified regime, cusp-like upwellings develop at the edges of large basins in the lower layer. Due to mixing, the volume of lower fluid decreases to zero over a finite time. Empirical scaling laws for the duration of mixing and for the peak temperature difference between the two fluids are derived and allow extrapolation to planetary mantles.

Effect of a Time-Dependent Stenosis on Flow Through a Tube

1968 ◽  
Vol 90 (2) ◽  
pp. 248-254 ◽  
Author(s):  
D. F. Young
Keyword(s):  
Stokes Equations ◽  
Flow Characteristics ◽  
Time Dependent ◽  
Arterial System ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Dependent Growth ◽  
Abnormal Tissue

A common occurrence in the arterial system is the narrowing of arteries due to the development of atherosclerotic plaques or other types of abnormal tissue development. As these growths project into the lumen of the artery, the flow is disturbed and there develops a potential coupling between the growth and the blood flow through the artery. A discussion of the various possible consequences of this interaction is given. It is noted that very small growths leading to mild stenotic obstructions, although not altering the gross flow characteristics significantly, may be important in triggering biological mechanisms such as intimal cell proliferation or changes in vessel caliber. An analysis of the effect of an axially symmetric, time-dependent growth into the lumen of a tube of constant cross section through which a Newtonian fluid is steadily flowing is presented. This analysis is based on a simplified model in which the convective acceleration terms in the Navier-Stokes equations are neglected. Effect of growth on pressure distribution and wall shearing stress is given and possible biological implications are discussed.

Sign in / Sign up

Export Citation Format

Share Document