scholarly journals Internally heated convection beneath a poor conductor

2015 ◽  
Vol 771 ◽  
pp. 36-56 ◽  
Author(s):  
David Goluskin
Keyword(s):  
Lower Bound ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Static State ◽  
Long Wavelength ◽  
Various Boundary Conditions ◽  
Mean Temperature ◽  
Long Wavelength Asymptotics ◽  
The Mean ◽  
Bounded Below

We consider convection in an internally heated (IH) layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable ($R_{L}$), and the energy analysis yields a Rayleigh number below which it is globally stable ($R_{E}$). For various boundary conditions on the velocity, exact expressions for $R_{L}$ and $R_{E}$ are found using long-wavelength asymptotics. Each $R_{E}$ is strictly smaller than the corresponding $R_{L}$ but is within 1 %. The lower bound on the mean temperature is proven for no-slip velocity boundary conditions using the background method. The bound guarantees that the mean temperature of the fluid, relative to that of the top boundary, grows with the heating rate ($H$) no slower than $H^{2/3}$.

Turbulent convection in a horizontal layer of water

1973 ◽  
Vol 60 (1) ◽  
pp. 141-159 ◽  
Author(s):  
T. Y. Chu ◽  
R. J. Goldstein
Keyword(s):  
Heat Transfer ◽  
Temperature Distribution ◽  
Rayleigh Number ◽  
Power Law ◽  
Bounding Surface ◽  
Interferometric Method ◽  
Number Range ◽  
Mean Temperature ◽  
The Mean ◽  
Rayleigh Numbers

Overall heat transfer and mean temperature distribution measurements have been made of turbulent thermal convection in horizontal water layers heated from below. The Nusselt number is found to be proportional to Ra0·278 in the range 2·76 × 105 < Ra < 1·05 × 108. Eight discrete heat flux transitions are found in this Rayleigh number range. An interferometric method is used to measure the mean temperature distribution for Rayleigh numbers between 3·11 × 105 and 1·86 × 107. Direct visual and photographic observations of the fluctuating interferogram patterns show that the main heat transfer mechanism is the release of thermals from the boundary layers. For relatively low Rayleigh numbers (up to 5 × 105) many of the thermals reach the opposite surface and coalesce to form large masses of relatively warm fluid near the cold surface and masses of cold fluid near the warm surface, resulting in a temperature-gradient reversal. With increasing Rayleigh numbers, fewer and fewer thermals reach the opposite bounding surface and the thermals show persistent horizontal movements near the bounding surfaces. The central region of the layer becomes an isothermal core. The mean temperature distributions for the high Rayleigh number range are found to follow a Z−2 power law over a considerable range, where Z is the distance from the bounding surface. A very limited agreement with the theoretically predicted Z−1 power law is also found.

Finite amplitude convection with changing mean temperature. Part 2. An experimental test of the theory

1968 ◽  
Vol 33 (3) ◽  
pp. 457-463 ◽  
Author(s):  
Ruby Krishnamurti
Keyword(s):  
Heat Flux ◽  
Fluid Layer ◽  
Vertical Direction ◽  
Finite Amplitude ◽  
Photographic Film ◽  
Static State ◽  
Mean Temperature ◽  
Constant Rate ◽  
The Mean ◽  
Rayleigh Numbers

It has been found in part 1 (Krishnamurti 1968) that when the mean temperature of a fluid layer is changing at a constant rate η, hexagonal flows are stable in a range of Rayleigh numbers near the critical. The direction of flow depends upon the sign of η. The static state is unstable to finite amplitude disturbances at Rayleigh numbers below the critical point predicted by linear theory.The validity of this theory is tested in an experiment in which the heat flux is measured as a function of η and Rayleigh number. The horizontal plan form is determined from the side by continuously exposing a photographic film moving in a vertical direction as tracers in different regions of the fluid are illuminated. Finite amplitude instability and hexagonal cells are indeed observed.

Finite amplitude convection with changing mean temperature. Part 1. Theory

1968 ◽  
Vol 33 (3) ◽  
pp. 445-455 ◽  
Author(s):  
Ruby Krishnamurti
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Stable Solution ◽  
Boussinesq Equations ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Temperature Fields ◽  
Physical Parameters ◽  
Mean Temperature ◽  
The Mean

When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude ε. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.

scholarly journals The Effect of Inclination on Natural Convective Heat Transfer from a Slender Cuboid

Processes ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1668
Author(s):  
Abdulrahim Kalendar ◽  
Yousuf Alhendal ◽  
Shafqat Hussain ◽  
Patrick Oosthuizen
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Vertical Axis ◽  
Surface Boundary ◽  
Nusselt Numbers ◽  
The Mean

A numerical study was undertaken of the naturally occurring laminar convective heat transfer from a slender cuboid with a relatively narrow cross-section (square) and an exposed top surface. The cuboid was perpendicularly placed on an adiabatic flat base plate and two types of surface boundary conditions were considered. The slender cuboid was inclined relative to the vertical axis at angles ranging from 0 to 180 degrees. The flow was considered symmetrical along the vertical axis of the slender cuboid. The equations governing the system were numerically solved in terms of dimensionless variables using FLUENT software. From the results obtained, mean Nusselt numbers over the slender cuboid were calculated using parameters such as: the Rayleigh number for heat flux, Ra*; the Rayleigh number, Ra; the slender cuboid dimensionless width, i.e., the ratio of the width to the height of the heated slender cuboid, W = w/h; and the position of the slender cuboid relative to the vertical, φ. Simulation results were produced for the boundary conditions of constant temperature, constant heat flux, and for Pr = 0.7. The effects of these parameters on the mean Nusselt number for the combined and for the individual surfaces of the slender cuboid are presented and the mean Nusselt numbers are correlated.

The Effect of Temperature Modulation on Natural Convection in a Horizontal Layer Heated From Below: High-Rayleigh-Number Experiments

1994 ◽  
Vol 116 (3) ◽  
pp. 614-620 ◽  
Author(s):  
J. Mantle ◽  
M. Kazmierczak ◽  
B. Hiawy
Keyword(s):  
Heat Transfer ◽  
Steady State ◽  
Rayleigh Number ◽  
Temperature Difference ◽  
Wall Temperature ◽  
Bottom Wall ◽  
Large Temperature ◽  
Temperature Modulation ◽  
Mean Temperature ◽  
The Mean

An experimental investigation was conducted to study the effects of wall temperature modulation in a horizontal fluid layer heated from below. A series of 45 transient experiments was performed in which the bottom wall temperature changed periodically with time in a “sawtoothlike” fashion. The amplitude of the bottom wall temperature oscillation varied from 3 to 70 percent of the enclosure’s mean temperature difference, and the period of the temperature swings ranged from 43 seconds to 93 minutes. With water as the fluid in the test cell, the flow was fully turbulent at all times. The Rayleigh number of the experiments (based on the enclosure’s height and on the mean temperature difference) was 0.4 × 108 < Ra < 1.2 × 109. It was found that for small changes in the bottom wall temperature, the cycle-averaged heat transfer through the layer was unchanged, independent of the period, and was equal in magnitude to the well-established steady-state value when the hot wall is evaluated at the mean temperature. However, this study shows that the cycle-averaged heat transfer increases notably, up to 12 percent as compared to the steady-state value, for the experiments with large temperature modulations. Futhermore, it was observed that the enchancement was a function of the amplitude and period of the oscillation.

On sideways diffusive instability

1971 ◽  
Vol 49 (2) ◽  
pp. 279-288 ◽  
Author(s):  
J. E. Hart
Keyword(s):  
Rayleigh Number ◽  
Boundary Conditions ◽  
Motion Field ◽  
Vertical Slot ◽  
Mean Fields ◽  
The Mean ◽  
Shear Instabilities ◽  
The Stability ◽  
Proper Account ◽  
Convective Rolls

We consider the stability of the motion generated in a differentially heated vertical slot filled with a linearly stratified salt solution. The theoretical mean motion field between infinite plates is a function of the Rayleigh number Rs, = gβ(∂S0/∂z) D4/k8ν. If Rs is zero the salinity does not enter the problem and one finds instability in the form of stationary rolls which obtain most of their energy from the basic velocity field. Even at very small Rs of order -1000 these shear instabilities are replaced by diffusively destabilized convective rolls which appear at a thermal Rayleigh number Ra = gαΔTD3/kTν which is two orders of magnitude less than that required for the shear generated modes. The present calculations, which take proper account of both the mean fields and the boundary conditions, give results which compare somewhat more favourably with the experimental results of Thorpe, Hutt & Soulsby (1969) than the theory put forward by these authors. It is shown why their theory, which deals with different boundary conditions from those in the experiment, gives adequate results as Rs tends to negative infinity.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

scholarly journals Horizontal convection in a rectangular enclosure driven by a linear temperature profile

2021 ◽  
Author(s):  
Tianyong Yang ◽  
Bofu Wang ◽  
Jianzhao Wu ◽  
Zhiming Lu ◽  
Quan Zhou
Keyword(s):  
Rayleigh Number ◽  
Temperature Profile ◽  
Scaling Law ◽  
Thermal Plume ◽  
Steady Regime ◽  
Linear Temperature ◽  
Square Enclosure ◽  
Horizontal Convection ◽  
Kinetic Boundary Layer ◽  
The Mean

AbstractThe horizontal convection in a square enclosure driven by a linear temperature profile along the bottom boundary is investigated numerically by using a finite difference method. The Prandtl number is fixed at 4.38, and the Rayleigh number Ra ranges from 107 to 1011. The convective flow is steady at a relatively low Rayleigh number, and no thermal plume is observed, whereas it transits to be unsteady when the Rayleigh number increases beyond the critical value. The scaling law for the Nusselt number Nu changes from Rossby’s scaling Nu ∼ Ra1/5 in a steady regime to Nu ∼ Ra1/4 in an unsteady regime, which agrees well with the theoretically predicted results. Accordingly, the Reynolds number Re scaling varies from Re ∼ Ra3/11 to Re ∼ Ra2/5. The investigation on the mean flows shows that the thermal and kinetic boundary layer thickness and the mean temperature in the bulk zone decrease with the increasing Ra. The intensity of fluctuating velocity increases with the increasing Ra.

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