Fixed Temperature Boundary Conditions Research Articles

Scaling laws for Rayleigh–Bénard convection between Navier-slip boundaries

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number ( ${\textit {Nu}}$ ). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number ( ${\textit {Pr}}$ ) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$ , ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$ , ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$ , ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$ .

Coupled thermo-mechanical analysis of creep in a rotating FGMEE annular plate under complex thermal loading considering solar radiation, convection, and internal heat source

In this analysis, the creep responses of a non-constant thickness annular plate was presented. The material of disc is assumed functionally graded magneto-electro-elastic (FGMEE) in which the material properties change through the radius. Also, the heat transfer coefficients for convection and conduction are functions of radius and temperature. At first, the equation of heat transfer accounting for thermal gradient, convection boundary conditions, internal heat generation, and solar radiation effects was derived. The differential transformation method (DTM) was used to solve the resulting nonlinear differential equation. The equilibrium equation for the annular plate including creep strain effects was then obtained. Ignoring creep strains, an analytical solution was obtained for the zero-time of this equation. Then, creep strains were introduced using Norton's law and the Prandtl-Reuss relations to find the stress and strain rates under fixed temperature boundary conditions. Next, the equation of strain rates including creep strains was solved analytically. Finally, an iterative approach was used to evaluate the time-dependent redistribution of creep stresses at any time point. Numerical examples highlighted the influences of key parameters like internal heat generation, convective heat transfer, grading index, solar radiation, thickness profile, and angular speed on the stresses, deformations and electric and magnetic potentials.

Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries

We consider two-dimensional Rayleigh–Bénard convection with Navier-slip and fixed temperature boundary conditions at the two horizontal rough walls described by the height function h. We prove rigorous upper bounds on the Nusselt number Nu which capture the dependence on the curvature of the boundary κ and the (non-constant) friction coefficient α explicitly. If h∈W2,∞ and κ satisfies a smallness condition with respect to α, we find Nu≲Ra12+∥κ∥∞, where Ra is the Rayleigh number, which agrees with the predicted Spiegel–Kraichnan scaling when κ = 0. This bound is obtained via local regularity estimates in a small strip at the boundary. When h∈W3,∞ , the functions κ and α are sufficiently small in L∞ and the Prandtl number Pr is sufficiently large, we prove upper bounds using the background field method, which interpolate between Ra12 and Ra512 with non-trivial dependence on α and κ. These bounds agree with the result in Drivas et al (2022 Phil. Trans. R. Soc. A 380 20210025) for flat boundaries and constant friction coefficient. Furthermore, in the regime Pr⩾Ra57 , we improve the Ra12 -upper bound, showing Nu≲α,κRa37, where ≲α,κ hides an additional dependency of the implicit constant on α and κ.

Effect of thermal boundary condition and turbulent models on the combustion simulation of ethylene-fueled scramjet combustor

Wall thermal boundary conditions and turbulent models can affect flow and combustion simulations but are seldom considered in the turbulent modeling of supersonic combustors. This work investigated the effect of thermal boundary conditions and four turbulent models on turbulent combustion in a cavity-stabilized scramjet combustor. Results showed that the thermal boundary condition had a noticeable influence on the temperature fields. Changing the thermal boundary condition from zero gradient to a fixed lower temperature considerably reduced the maximum temperature but did not affect the temperature distribution. The fixed temperature boundary condition generated a slightly larger reaction heat release near the upper region of the cavity. However, the mass fraction of carbon dioxide was low for a fixed low temperature. The pressure increased near the rear of the cavity but decreased elsewhere at a fixed temperature. Reynolds-averaged models (k-epsilon, k-omega, and realizable k-epsilon) tend to over-predict the temperature and turbulent kinetic energy but under-predict the mass fraction of carbon dioxide. The detached Eddy simulation also under-predicts carbon dioxide but predicts a more accurate temperature.

Instabilities of rotating inclined buoyancy layers.

The boundary layer near a cooled inclined plate, which is immersed in a stably stratified fluid rotating about an axis parallel to the direction of gravity, is a model for katabatic flows at high latitudes. In this paper the base flow of such an inclined buoyancy layer is solved analytically for arbitrary Prandtl numbers. By applying linear stability analyses, five unstable modes are identified for both the fixed temperature and the isoflux boundary conditions, i.e., the stationary longitudinal roll (LR) mode, the oblique roll with low streamwise wave-number (OR-1) and high streamwise wave-number (OR-2) modes, and the Tolmien-Schlichting (TS) wave with low streamwise wave-number (TS-1) and high streamwise wave-number (TS-2) modes. It is indicated that the Coriolis effect induced by the rotation leads the critical modes to be three dimensional, and a larger tilt angle of the plate and stronger Coriolis effect cause both TS wave modes to be more unstable for both thermal boundary conditions. When the Coriolis effect is considered, the OR-1 and OR-2 modes are the most unstable mode at low and high tilt angles, respectively, but the TS-1 wave mode may be the most unstable one when the plate is nearly vertical. In addition, the spanwise phase velocities of the TS wave modes change directions as the tilt angle passes some threshold values for both thermal boundary conditions except for the TS-1 wave mode with a fixed temperature boundary condition, which propagates in the same spanwise direction for all explored tilt angles.

Development of a hybrid model for fuel performance analysis of spherical fuel elements under normal conditions

The spherical fuel element (SFE) dispersed with many tri-isotropic fuel (TRISO) particles has been primarily employed in the high-temperature gas-cooled reactor. Owing to the complex structure of SFE, behaviors will be evolved for both TRISO particles and the fuel matrix with the increasing burnup. To evaluate the fuel performance of SFE, the homogenization method has been conventionally adopted for SFE scale modeling, and dispersed TRISO particles are analyzed subsequently with fixed temperature boundary conditions. However, owing to the varying properties of TRISO particles under reactor operation, the homogenization models dedicated to certain circumstances can hardly reflect the properties evolution of TRISO with high fidelity, which may lead to uncertainties in analysis results. To this end, a hybrid model, namely the conventional homogenization model informed by the lower scale model online, has been developed in this paper, which may increase modeling fidelity for both SFE and particles. Specifically, property models for the TRISO particle and graphite matrix have been implemented in COMSOL. Subsequently, based on the linking of COMSOL and MATLAB, the hybrid model for SFE analysis was established. Through the hybrid model, the temperature field and stress distribution of TRISO, the evolution of effective properties, and the thermal and mechanical state of the SFE were obtained and analyzed. The results show that within the scope of this paper, the central temperature of TRISO in the innermost region rises with the increasing burnup for the decreasing thermal conductivity of materials, and the peak value is about 1324 K, which is about 60 K higher than that in the outermost region. The difference in the maximum plenum pressure is more than 5 MPa among TRISO particles, which is mainly caused by the variations in CO production. Consequently, the hoop stress of SiC in the outermost region is slightly elevated. The distribution of effective thermal conductivity (ETC) for TRISO is nearly 4.3 W/(m·K) at the beginning of life (BOL), while it was reduced to 3.6 W/(m·K) at the end of life (EOL). There are marginal variations along the radial direction for the ETC of TRISO. Meanwhile, the ETC of SFE decreases from about 21 W/(m·K) for fresh fuel to 13 W/(m·K) for spent fuel. The predicted effective elastic modulus and coefficient of thermal expansion of SFE increase gradually versus time. The former is caused by the densification of pyrolytic carbon, and the latter is mainly determined by the high volume fraction of matrix and the changes of CTE for pyrolytic carbon. In addition, the comparison in the temperature field between the proposed model and the Differential Effective Material Theory (D-EMT) model may further demonstrate the feasibility and effectiveness of the hybrid model.

Onset of Convection in Rotating Spherical Shells: Variations With Radius Ratio

AbstractConvection in rotating spherical layers of fluid is ubiquitous in spherical astrophysical objects like planets and stars. A complete understanding of the magnetohydrodynamics requires understanding of the linear problem—when convection onsets in these systems. This is a fluid dynamics problem that has been studied since the early 1900s. Theoretical scaling laws exist for the variation of critical quantities—the Rayleigh number Rac, the azimuthal wavenumber mc, and the angular drift frequency ωc—with respect to the Ekman number E. However, their variation with the radius ratio χ of the spherical shell is still poorly studied. To address this, we use an open source eigenvalue code Kore to compute these critical quantities over an extensive range of parameters spanning four decades in Ekman number and a dense grid of radius ratio from very thick to very thin shells, focusing on no‐slip and fixed temperature boundary conditions. We find that these variations are explained well by the theoretical scaling laws, especially at low E, but variations in radius ratio also exist. We obtain scaling laws of boundary layer thicknesses and spatial extent of onset modes with respect to the Ekman number which differ only slightly from theoretical scalings. We show that our data set can be used to obtain good estimates of critical quantities in the moderate E range, where the vast majority of current geophysical and astrophysical fluid dynamics simulations are performed, yet where asymptotic theory is only moderately accurate. We further verify asymptotic predictions and determine best‐fit asymptotic model coefficients.

Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries.

We study two-dimensional Rayleigh-Bénard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number [Formula: see text], this estimate interpolates between the Whitehead-Doering bound by [Formula: see text] for free-slip conditions (Whitehead & Doering. 2011 Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106, 244501) and the classical Doering-Constantin [Formula: see text] bound (Doering & Constantin. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957-5981). This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

CFD analysis for heat transfer comparison in circular, rectangular and elliptical tube heat exchangers filled with PCM

A two-dimensional computational fluid dynamics (CFD) investigation of the melting of phase change material (PCM) and heat transfer characteristics of a heat exchanger of shell and tube type having circular, rectangular and elliptical tubes are analyzed in this study. Gallium in the rectangular shell of heat exchange is considered as the working fluid in this present study. A fixed temperature boundary condition is applied at the circumference of the tubes and a standard k-epsilon turbulence model is used for the simulation. The variation in the shape of the tubes at a fixed temperature of 60 °C has been utilized for the analysis and the effect of the shape of these tubes on melting of PCM in terms of liquid proportion and melting time has been analyzed. The melting duration of PCM is 745 s for a configuration with circular tubes, 650 s for a configuration with rectangular tubes, and 740 s for a configuration with elliptical tubes and the duration by the PCM to reach 60 °C for a configuration with circular tubes is 800 s, for a configuration with rectangular tubes is 725 s, and for a configuration with elliptical tubes is 770 s. Hence it is indicated by the results that the lowest melting time of PCM can be obtained by the heat exchanger having rectangular tubes followed by elliptical tubes and circular tubes at a temperature of 60 °C.

The impact of time‐varying sea surface temperature on UK regional atmosphere forecasts

AbstractA new approach to improve the ocean surface boundary condition used in regional numerical weather prediction is proposed. Typically, regional atmosphere forecast systems assume a fixed sea surface temperature during a simulation. The study assesses the use of ocean temperature from an operational regional ocean model as an evolving lower boundary in a kilometre‐scale regional atmosphere configuration centred on the UK. Simulations of a winter and two five day duration summer case studies associated with anomalously warm temperatures are considered. The largest impact is found in summer, when a growing cold bias in mean temperature over land compared with observations is apparent when using a fixed global‐scale analysis lower boundary condition. The mean error is improved by 0.1 K when using a fixed temperature boundary condition from a kilometre‐scale regional ocean model initial condition. When using hourly surface temperature data from the same regional ocean model, the error is improved by 0.5 K for this case. Prediction of daytime maximum air temperature is also improved during the summer heat wave cases. A winter case study shows marginal improvement over the ocean and negligible changes over land. These results are confirmed in longer duration experiments using an hourly cycling regional forecast system with data assimilation for summer and winter periods. A systematic and statistically significant improvement of near‐surface temperature verification relative to the observations over land is demonstrated for both summer and winter using the new approach. This study supports future operational implementation of a time‐varying lower boundary for regional numerical weather prediction.

The Influence of Leg Shape on Thermoelectric Performance Under Constant Temperature and Heat Flux Boundary Conditions

Global energy resource consumption is increasing rapidly, and wasted energy (the portion not used for useful services) comprises a significant fraction of the energy consumed. Much of the wasted energy is in the form of heat. Thermoelectric devices offer the potential to convert this wasted heat into electricity, and they can be used for thermal management by pumping heat. The shape of thermoelectric devices, particularly the active semiconducting material units (or thermoelectric legs) within the device, has remained largely unchanged. However, the advent of new manufacturing techniques such as additive manufacturing enables customizable device shapes. Given these new manufacturing capabilities, exploring atypical, or non-rectangular, leg geometries becomes increasingly relevant. This work focuses on understanding the effect of different thermoelectric leg designs on thermoelectric device performance. Various leg shapes were studied for their thermal and electrical performance under different thermal boundary conditions. The shapes studied include rectangular prisms, rectangular prisms with interior hollows, trapezoids, hourglass, and Y-shapes. Temperature gradients and the electrical potentials of the thermoelectric legs are determined using the COMSOL Multiphysics Thermoelectric Module. Both fixed temperature and fixed heat flux boundary conditions were examined numerically. Among the geometries investigated, the hourglass-shaped thermoelectric leg, subjected to a fixed temperature boundary condition, is found to have the best thermal and electrical performance. The hourglass-shaped leg results in more than double the electrical potential and maximum power compared to the conventional rectangular shape when the cold side experiences a natural convection boundary condition. Under a fixed heat flux boundary condition on the hot side, a trapezoid-shaped leg results in almost double the electrical potential and a 50% increase in the power output compared to the conventional leg shape. Varying boundary conditions, which reflect different device operating conditions, result in different performance values for the same leg shapes. These findings underscore the importance of leg geometry on electrical and thermal performance of a thermoelectric leg, as well as the importance of considering the device operating condition when selecting the best leg shape.

Modelling Heat Loss Effects in the Large Eddy Simulation of a Lean Swirl-Stabilised Flame

The flame in a gas turbine model combustor close to blow-off is studied using large eddy simulation with the objective of investigating the sensitivity of including different heat loss effects within the modelling. A presumed joint probability density function approach based on the mixture fraction and progress variable with unstrained flamelets is used. The normalised enthalpy is included in the probability density function to account for heat loss within the flame. Two simulations are presented that use fixed temperature boundary conditions, and use adiabatic and non-adiabatic formulations of the combustion model. The results are compared against the previous fully adiabatic case and experimental data. The statistics for the simulations are similar to the results obtained from the fully adiabatic case. Improved statistics are obtained for the temperature in the near-wall regions. The non-adiabatic flamelet case shows the average reaction rate values at the flame root are approximately 50% smaller in comparison to the adiabatic flamelet cases. This causes the lift-off height to be overestimated. The time series of the lift-off height and the volume integrated heat release rate show that including non-adiabatic flamelets causes the flame to be highly unstable. A higher enthalpy deficit is seen in the near-field regions when the flame root is not present and experiencing some lift-off, suggesting that the flame is more dynamic when including heat loss.

The atmospheric Rayleigh-Bénard problem on the f-plane

When applied to a system of sizeable vertical extent that can undergo adiabatic expansion/compression, the Rayleigh-Bénard treatment of convection between two parallel plates, kept at constant temperature, needs to be amended with the consideration of potential temperature as the conserved thermodynamic variable. The fixed-temperature boundary conditions are therefore expressed as a combination of potential temperature and pressure, and this causes the solutions to be a mixture of the odd and even modes of the classical problem. Here, solutions are presented for a rotating system, which supports both stationary and oscillatory modes. While the stationary modes are all stabilized by this mechanism, as was shown previously for a nonrotating system, the oscillatory modes can have a lower critical Rayleigh number than their traditional counterpart, when the Prandtl number is approximately between 0.2 and 1.0.

An inverse optimization of convection heat transfer in rectangle channels with ribbed surface based on the extremum principle of entransy dissipation

The ribbed surface is a widely used convection heat transfer enhancement technology. The geometric parameters of the ribbed surface have great influences on its heat transfer performance. It is crucial to determine the optimum geometric parameters. In the present work, the optimal arrangement of ribs mounted in a rectangular channel is provided by solving the inverse optimization problem with the entransy dissipation as the objective function. The parameter analysis based on the direct solution is firstly carried out to get an approximate range near the optimal pitch ratio which can provide the initial value for the inverse optimization problem; afterwards, the inverse optimization problem is solved by a modified simplified conjugate gradient method, and the accurate optimal pitch ratio is obtained; finally the optimization results are discussed in detail. The results indicate that, (1) the inverse optimization analysis presented in this paper can provide an accurate optimal value of the pitch ratio; (2) taking the entransy dissipation as the objective function could lead to a faster converge process and give a more stable result than taking the temperature data as the objective function; (3) the optimal value under the fixed heat flux boundary condition is slightly larger than that under the fixed temperature boundary condition.

Weakly non-Boussinesq convection in a gaseous spherical shell.

We examine the dynamics associated with weakly compressible convection in a spherical shell by running 3D direct numerical simulations using the Boussinesq formalism [Spiegel and Veronis, Astrophys. J. 131, 442 (1960)AJLEEY0004-637X10.1086/146849]. Motivated by problems in astrophysics, we assume the existence of a finite adiabatic temperature gradient ∇T_{ad} and use mixed boundary conditions for the temperature with fixed flux at the inner boundary and fixed temperature at the outer boundary. This setup is intrinsically more asymmetric than the more standard case of Rayleigh-Bénard convection in liquids between parallel plates with fixed temperature boundary conditions. Conditions where there is substantial asymmetry can cause a dramatic change in the nature of convection and we demonstrate that this is the case here. The flows can become pressure- rather than buoyancy-dominated, leading to anomalous heat transport by upflows. Counterintuitively, the background temperature gradient ∇T[over ¯] can develop a subadiabatic layer (where g·∇T[over ¯]<g·∇T_{ad}, where g is gravity) although convection remains vigorous at every point across the shell. This indicates a high degree of nonlocality.

Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection

Multiple authors have claimed that the natural convection of a fluid is a process that exhibits maximum entropy production (MEP). However, almost all such investigations were limited to fixed temperature boundary conditions (BCs). It was found that under those conditions, the system tends to maximize its heat flux, and hence it was concluded that the MEP state is a dynamical attractor. However, since entropy production varies with heat flux and difference of inverse temperature, it is essential that any complete investigation of entropy production allows for variations in heat flux and temperature difference. Only then can we legitimately assess whether the MEP state is the most attractive. Our previous work made use of negative feedback BCs to explore this possibility. We found that the steady state of the system was far from the MEP state. For any system, entropy production can only be maximized subject to a finite set of physical and material constraints. In the case of our previous work, it was possible that the adopted set of fluid parameters were constraining the system in such a way that it was entirely prevented from reaching the MEP state. Hence, in the present work, we used a different set of boundary parameters, such that the steady states of the system were in the local vicinity of the MEP state. If MEP was indeed an attractor, relaxing those constraints of our previous work should have caused a discrete perturbation to the surface of steady state heat flux values near the value corresponding to MEP. We found no such perturbation, and hence no discernible attraction to the MEP state. Furthermore, systems with fixed flux BCs actually minimize their entropy production (relative to the alternative stable state, that of pure diffusive heat transport). This leads us to conclude that the principle of MEP is not an accurate indicator of which stable steady state a convective system will adopt. However, for all BCs considered, the quotient of heat flux and temperature difference F / Δ T —which is proportional to the dimensionless Nusselt number—does appear to be maximized.

A mathematical model for nanoparticle melting with size-dependent latent heat and melt temperature

In this paper, we study the melting of a spherical nanoparticle. The model differs from previous ones in that a number of features have been incorporated to match experimental observations. These include the size dependence of the latent heat and a cooling condition at the boundary (as opposed to the fixed temperature condition used in previous studies). Melt temperature variation and density change are also included. The density variation drives the flow of the outer fluid layer. The latent heat variation is modelled by a new relation, which matches experimental data better than previous models. A novel form of Stefan condition is used to determine the position of the melt front. This condition takes into account the latent heat variation, the energy required to create new surface and the kinetic energy of the displaced fluid layer. Results show that melting times can be significantly faster than predicted by previous theoretical models; for smaller particles, this can be around a factor 3. This is primarily due to the latent heat variation. The previously used fixed temperature boundary condition had two opposing effects on melt times: the implied infinite heat transfer led to faster melting but also artificially magnified the effect of kinetic energy, which slowed down the process. We conclude that any future models of nanoparticle melting must be based on the new Stefan condition and account for latent heat variation.

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

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.

Cooling Capacity Figure of Merit for Phase Change Materials

In this paper, a figure of merit for the cooling capacity (FOMq) of phase change materials (PCMs) is defined from the analytical solution of the two-phase Neumann–Stefan problem of melting of a semi-infinite material with a fixed temperature boundary condition (BC). This figure of merit is a function of the thermophysical properties of a PCM and is proportional to the heat transfer across the interface with the surrounding medium in this general case. Thus, it has important implications for design and optimization of PCMs for high heat-flux thermal management applications. FOMq of example low melting point metals are presented which exceed those in common nonmetallic PCMs over the same temperature range by over an order of magnitude.

FULLY CONVECTIVE MAGNETOROTATIONAL TURBULENCE IN STRATIFIED SHEARING BOXES

We present a numerical study of turbulence and dynamo action in stratified shearing boxes with zero magnetic flux. We assume that the fluid obeys the perfect gas law and has finite (constant) thermal diffusivity. We choose radiative boundary conditions at the vertical boundaries in which the heat flux is propor- tional to the fourth power of the temperature. We compare the results with the corresponding cases in which fixed temperature boundary conditions are applied. The most notable result is that the formation of a fully convective state in which the density is nearly constant as a function of height and the heat is transported to the upper and lower boundaries by overturning motions is robust and persists even in cases with radiative boundary conditions. Interestingly, in the convective regime, although the diffusive transport is negligible the mean stratification does not relax to an adiabatic state.