scholarly journals The asymptotic equivalence of fixed heat flux and fixed temperature thermal boundary conditions for rapidly rotating convection

2015 ◽  
Vol 784 ◽  
Author(s):  
Michael A. Calkins ◽  
Kevin Hale ◽  
Keith Julien ◽  
David Nieves ◽  
Derek Driggs ◽  
...  
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Layer Structure ◽  
Plane Layer ◽  
Order System ◽  
Leading Order ◽  
Fixed Temperature ◽  
Rotating Convection ◽  
Thermal Boundary Conditions

The influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is investigated for the case of stress-free mechanical boundary conditions. It is shown that whereas the leading-order system satisfies fixed temperature boundary conditions implicitly, a double boundary layer structure is necessary to satisfy the fixed heat flux thermal boundary conditions. The boundary layers consist of a classical Ekman layer adjacent to the solid boundaries that adjust viscous stresses to zero, and a layer in thermal wind balance just outside the Ekman layers that adjusts the normal derivative of the temperature fluctuation to zero. The influence of these boundary layers on the interior geostrophically balanced convection is shown to be asymptotically weak, however. Upon defining a simple rescaling of the thermal variables, the leading-order reduced system of governing equations is therefore equivalent for both boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading-order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

Effects of thermal boundary conditions on rotating Rayleigh-Bénard convection with implications on geophysical and astrophysical systems

2021 ◽  
Author(s):  
Janet Peifer ◽  
Onno Bokhove ◽  
Steve Tobias
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Biot Number ◽  
Thermal Flux ◽  
Thermal Boundary Condition ◽  
Actual Condition ◽  
Fixed Temperature ◽  
Thermal Boundary Conditions ◽  
Core Dynamics

<p>Rayleigh-Bénard convection (RBC) is a fluid phenomenon that has been studied for over a century because of its utility in simplifying very complex physical systems. Many geophysical and astrophysical systems, including planetary core dynamics and components of weather prediction, are modeled by including rotational forcing in classic RBC. Our understanding of these systems is confined by experimental and numerical limits, as well as theoretical assumptions. </p><p>The role of thermal boundary condition choice on experimental studies of geophysical and astrophysical systems has been often been overlooked, which could account for some lack of agreement between experimental and numerical models as well as the actual flows. The typical thermal boundary conditions prescribed at the top and the bottom of a convection system are fixed temperature conditions, despite few real geophysical systems being bounded with a fixed temperature. A constant heat flux is generally more applicable for real large-scale geophysical systems. However, when this condition is applied in numerical systems, the lack of fixed temperature can cause a temperature drift. In this study, we seek to minimize temperature drifting by applying a fixed temperature condition on one boundary and a fixed thermal flux on the other.</p><p>Experimental boundary conditions are also often assumed to be a fixed temperature. However, the actual condition is determined by the ratio of the height and thermal conductivity of the boundary material to that of the contained fluid, known as the Biot number. The relationship between the Biot number and thermal boundary condition behavior is defined by the Robin, or 'thin-lid', boundary condition such that low Biot number boundaries are essentially fixed thermal flux and high Biot number boundaries are essentially fixed temperature. </p><p>This study seeks to strengthen the link between numerical and experimental models and geophysical flows by investigating the effects of thermal boundary conditions and their relationship to real-world processes. Both fixed temperature and fixed flux boundary conditions are considered. In addition, the Robin boundary condition is studied at a range of Biot numbers spanning from fixed temperature to fixed flux, allowing intermediate conditions to be investigated. Each system is studied at increasingly rapid rotation rates, corresponding to decreasing Ekman numbers as low as Ek=10<sup>-5</sup> Heat transport is analyzed using the Nusselt number, Nu, and the form of the solution is described by the number of convection rolls and time-dependency. Further investigations will analyze Nu and fluid movement within a system with heterogeneous heat flux condition on the  sidewall boundary conditions, which is useful in the study of planetary core dynamics. The results of this study have implications for improvements in modeling geophysical systems both experimentally and numerically. </p>

Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal, and Concave Parabolic Profiles

2006 ◽  
Author(s):  
A. Aziz
Keyword(s):  
Thermal Analysis ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Fixed Temperature ◽  
Thermal Boundary Conditions ◽  
Convective Fin ◽  
The Relationship ◽  
Alternative Solutions ◽  
Dimensionless Heat Flux ◽  
Dimensionless Heat

The traditional thermal analysis of fins is based on the assumption of specified thermal boundary conditions at the base and tip of the fin. For situations when the fin base is in contact with a fluid experiencing condensation and the fin is required to remove the energy released by the fluid, the base is subjected to two boundary conditions: a fixed temperature and a fixed heat flux. This paper develops solutions for the temperature distribution in the fins under these conditions. Solutions are provided for rectangular, trapezoidal, and concave parabolic (finite tip thickness). Results illustrating the relationship between the dimensionless heat flux, the fin parameter, and dimensionless tip temperature are provided for all three geometries. The case of convective fin tip is also considered and lead to a relationship between the dimensionless heat flux, the fin parameter, and the Biot number at the tip. The results presented here provide tools that not only complement the traditional analyses but are believed to have more direct relevance for fin designers.

scholarly journals Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions

2017 ◽  
Vol 828 ◽  
pp. 601-629 ◽  
Author(s):  
Jon E. Mound ◽  
Christopher J. Davies
Keyword(s):  
Heat Flux ◽  
Outer Boundary ◽  
Extreme Case ◽  
Creeping Flow ◽  
Supercritical Regime ◽  
Rotating Convection ◽  
Thermal Boundary Conditions ◽  
Core Convection ◽  
Rayleigh Numbers ◽  
Lateral Variations

Convection in the metallic cores of terrestrial planets is likely to be subjected to lateral variations in heat flux through the outer boundary imposed by creeping flow in the overlying silicate mantles. Boundary anomalies can significantly influence global diagnostics of core convection when the Rayleigh number, $Ra$, is weakly supercritical; however, little is known about the strongly supercritical regime appropriate for planets. We perform numerical simulations of rapidly rotating convection in a spherical shell geometry and impose two patterns of boundary heat flow heterogeneity: a hemispherical $Y_{1}^{1}$ spherical harmonic pattern; and one derived from seismic tomography of the Earth’s lower mantle. We consider Ekman numbers $10^{-4}\leqslant E\leqslant 10^{-6}$, flux-based Rayleigh numbers up to ${\sim}800$ times critical, and a Prandtl number of unity. The amplitude of the lateral variation in heat flux is characterised by $q_{L}^{\ast }=0$, 2.3, 5.0, the peak-to-peak amplitude of the outer boundary heat flux divided by its mean. We find that the Nusselt number, $Nu$, can be increased by up to ${\sim}25\,\%$ relative to the equivalent homogeneous case due to boundary-induced correlations between the radial velocity and temperature anomalies near the top of the shell. The $Nu$ enhancement tends to become greater as the amplitude and length scale of the boundary heterogeneity are increased and as the system becomes more supercritical. This $Ra$ dependence can steepen the $Nu\propto Ra^{\unicode[STIX]{x1D6FE}}$ scaling in the rotationally dominated regime, with $\unicode[STIX]{x1D6FE}$ for our most extreme case approximately 20 % greater than the equivalent homogeneous scaling. Therefore, it may be important to consider boundary heterogeneity when extrapolating numerical results to planetary conditions.

Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile

1964 ◽  
Vol 18 (4) ◽  
pp. 513-528 ◽  
Author(s):  
E. M. Sparrow ◽  
R. J. Goldstein ◽  
V. K. Jonsson
Keyword(s):  
Heat Flux ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Temperature Profile ◽  
Fluid Layer ◽  
Bounding Surface ◽  
Fixed Temperature ◽  
Linear Temperature ◽  
Lower Bounding ◽  
Horizontal Fluid Layer

An investigation is carried out to determine the conditions marking the onset of convective motion in a horizontal fluid layer in which a negative temperature gradient occurs somewhere within the layer. In such cases, fluid of greater density is situated above fluid of lesser density. Consideration is given to a variety of thermal and hydrodynamic boundary conditions at the surfaces which bound the fluid layer. The thermal conditions include fixed temperature and fixed heat flux at the lower bounding surface, and a general convective-radiative exchange at the upper surface which includes fixed temperature and fixed heat flux as special cases. The hydrodynamic boundary conditions include both rigid and free upper surfaces with a rigid lower bounding surface. It is found that the Rayleigh number marking the onset of motion is greatest for the boundary condition of fixed temperature and decreases monotonically as the condition of fixed heat flux is approached. Non-linear temperature distributions in the fluid layer may result from internal heat generation. With increasing departures from the linear temperature profile, it is found that the fluid layer becomes more prone to instability, that is, the critical Rayleigh number decreases.

Effects of Nanoparticle Migration on Water/Alumina Nanofluid Flow Inside a Horizontal Annulus with a Moving Core

2014 ◽  
Vol 31 (3) ◽  
pp. 291-305 ◽  
Author(s):  
A. Malvandi ◽  
D. D. Ganji
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Outer Wall ◽  
Constant Heat Flux ◽  
Constant Heat ◽  
Thermal Boundary Conditions ◽  
Alumina Nanofluid

AbstractThe present study is a theoretical investigation of the laminar flow and convective heat transfer of water/alumina nanofluid inside a horizontal annulus with a streamwise moving inner cylinder. A modified, two-component, four-equation, nonhomogeneous equilibrium model is employed for the alumina/water nanofluid, which fully accounts for the effect of the nanoparticle volume fraction distribution. To determine the effects of thermal boundary conditions on the migration of the nanoparticles, two cases are considered: constant heat flux at the outer wall with an adiabatic inner wall (Case A) and constant heat flux at the inner wall with an adiabatic outer wall (Case B). The numerical results indicate that the thermal boundary conditions at the pipe walls significantly affect the nanoparticle distribution, particularly in cases where the ratio of Brownian motion to thermophoretic diffusivities is small. Moreover, increasing the velocity of the moving inner cylinder reduces the heat transfer rate for Case A. Conversely, in Case B, the movement of the inner cylinder enhances the heat transfer rate, and anomalous heat transfer enhancement occurs when the thermophoretic force is dominant (in larger nanoparticles).

Finite amplitude convection cells

1967 ◽  
Vol 30 (3) ◽  
pp. 577-600 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Cross Section ◽  
Circular Cross Section ◽  
Maximum Heat ◽  
Mean Field Equations

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

Alternative and Relevant Representation to Heat Transfer Coefficient for Modeling the Heat Transfer Between a Fluid and a Non-Isothermal Wall in Transient Regime

2010 ◽  
Author(s):  
Benjamin Remy ◽  
Alain Degiovanni
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Transfer Coefficient ◽  
Steady State ◽  
Boundary Conditions ◽  
Transfer Coefficient ◽  
Flat Plate ◽  
Temperature Difference ◽  
Interfacial Heat Transfer ◽  
Thermal Boundary Conditions

This paper deals with the relevant model that can be proposed for modeling the interfacial heat transfer between a fluid and a wall in the case of space and time varying thermal boundary conditions. Usually, for a constant and uniform heat transfer (unidirectional steady-state regime), the problem can be solved introducing a heat transfer coefficient h, uniform in space and constant in time that linearly links the surface heat flux and the temperature difference between the wall temperature Tw and an equivalent fluid temperature Tf. The problem we consider in this work concerns the heat transfer between a steady-state fluid flow and a wall submitted to a transient and non uniform thermal solicitations, as for instance a steady-state flow on a flat plate submitted to a transient and space reduced heat flux. We will show that the more interesting representation for describing the interfacial heat transfer is not to define as usually done a non-uniform and variable heat transfer coefficient h(x,t) because as it depends on the thermal boundary conditions, it is not really intrinsic. We propose an alternative approach, which consists in introducing a generalized impedance Z(ω,p) that links in space and time domain the heat flux and the temperature difference through a double convolution product instead of a scalar product. After the presentation of the general problem, the simple case of a stationary piston flow that can be solved analytically will be considered for validation both in thermal steady-state and transient regimes. To conclude and show the interest of our approach, a comparison between a global approach and a numerical simulation in a more complex and realistic case taking into account the thermal coupling with a flat plate will be presented.

Steady Thermochromic Liquid Crystal Technique for Study of Conjugate Heat Transfer Problems

2003 ◽  
Author(s):  
C. J. Douglass ◽  
J. S. Kapat ◽  
E. Divo ◽  
A. J. Kassab ◽  
J. Tapley ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Liquid Crystal ◽  
Boundary Conditions ◽  
Surface Heat Flux ◽  
Conjugate Heat Transfer ◽  
Surface Heat ◽  
Test Block ◽  
Thermochromic Liquid Crystal ◽  
Thermal Boundary Conditions

This paper presents a steady measurement technique based on thermochromic liquid crystals (TLC) that can be used for study of conjugate heat transfer. In contrast to the more commonly used transient thermochromic liquid crystal technique, this technique requires steady-state experiments, and eliminates some of the limitations of the transient version at the cost of measurements or knowledge of thermal conditions all surfaces and increased computations for data reduction. This technique requires that thermal boundary conditions be known or measured on all internal or external surfaces of the test block. All surfaces that are exposed to external air flow are coated with a broad-bandwidth TLC. The thermal boundary conditions are then sent to a steady conduction solver that involves the boundary element method (BEM) and an inverse problem approach (BEM/IP). This combined BEM/IP approach minimizes the effects of random experimental error in measured data and calculates surface heat flux, from which the intended convective heat flux coefficients can then be calculated. The technique is applied to a prismatic stainless steel block exposed to warm air flows on three sides — an arrangement that has been used often to simulate flow through a blade tip gap. It is found that an in-situ pixel-by-pixel calibration of TLC hue vs temperature is needed in order to obtain reasonable accuracy. A calibration-curve-fit uncertainty of better than 0.4°C (at 95% confidence level) was obtained in this process. In the actual experiments, conjugate heat transfer was set up by passing cold water through three cooling channels that span the test block. Once the experiments are completed and the TLC colors are converted to surface temperature distributions, the BEM/IP approach is used to obtain surface heat flux distributions, and then distribution of heat transfer coefficients.

scholarly journals Finite element analysis of a composite piston for a diesel aircraft engine

2019 ◽  
Vol 179 (4) ◽  
pp. 107-111
Author(s):  
Konrad PIETRYKOWSKI ◽  
Paweł MAGRYTA ◽  
Krzysztof SKIBA
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Mechanical Load ◽  
Aircraft Engine ◽  
Element Analysis ◽  
Mechanical Loads ◽  
Thermal Boundary Conditions ◽  
Piston Crown ◽  
Geometrical Dimensions ◽  
Abaqus Software

The article presents calculations of thermal and mechanical loads of the piston, consisting of two parts: steel and aluminum. The calculations were made using FEM in the Abaqus software. The piston is characterized by a split construction and was equipped with a cooling oil channel. The piston will be used in an aircraft diesel engine characterized by opposite piston movement. The presented geometry of the piston is the next of the ones being developed earlier and contains preliminary assumptions as to the size and main geometrical dimensions. The thermal boundary conditions of the simulation tests assumed defined areas of heat reception surface and heating of the piston by defining a temperature map on its crown. The results of these studies were presented in the form of temperature distribution and heat flux on the surface of the tested element. The strength boundary conditions assumed a mechanical load in the form of pressure resulting from the pressure in the combustion chamber applied to the piston crown surface and the opposite pressure defined on the support at the surface of contact between the piston and the piston pin. The results of these tests were presented in the form of stress distribution on the surface of the tested element. As a result of the analyses carried out, the results constituting the basis for further modernization of the piston geometry were obtained.

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