The role played by patterned heating in reducing pressure losses within vertical conduits is investigated. The heating generates flow separation structures which reduce the direct contact between the stream and the sidewalls, thereby limiting the frictional resistance. This also modifies the temperature field thereby inducing a net buoyancy force which may either assist or oppose the pressure gradient required to maintain a fixed flow rate. If the flow Reynolds number is increased sufficiently, the separation structures may be washed away, which means that the pressure-gradient-reducing mechanism is eliminated. The details of the system response are a function of the form of spatial heating distribution, its intensity, the flow Reynolds number and the fluid Prandtl number. Carefully chosen heating of the two walls can induce a pattern interaction effect and a judicious choice of the two patterns can have as much as an order of magnitude effect on the system response.

This study presents a buoyancy-driven stability analysis in a three-dimensional inclined porous medium with a capillary transition zone that is formed between a non-wetting and an underlying wetting phase. In this two-phase, two-component, partially miscible system, a solute from a non-wetting phase diffuses into a porous layer saturated with a wetting-phase fluid, creating a dense diffusive boundary layer beneath an established capillary transition zone. Transient concentration and gravity-driven velocity fields are derived for the wetting phase while the saturation field remains fixed. Linear stability analysis with the quasi-steady-state approximation is employed to determine the onset of solutal convective instability for buoyancy-dominant, in-transition and capillary-dominant systems. The analysis of the problem leads to a differential eigenvalue problem composed of a system of three complex-valued equations that are numerically solved to determine the critical times, critical wavenumbers and neutral stability curves as a function of inclination angle for different Bond numbers. The layer inclination is shown to play an essential role in the stability of the problem, where the gravity-driven flow removes solute concentrations in the diffusive boundary layer. The results indicate that the horizontal porous layer exhibits the fastest onset of instability, and longitudinal rolls are always more unstable than oblique and transverse rolls. The inclination angle has a more substantial impact on stabilizing the diffusive boundary layer in the buoyancy-dominant than in the capillary-dominant systems. Furthermore, for both buoyancy-dominant and capillary-dominant systems, the critical times and wavenumbers vary exponentially with inclination angle ≤ 60° and follow the Stirling model.

Modern-day ocean circulation behaves as a complex forced convective system that is characterized by the decrease in water temperature but increase in water density with depth. The dissolved oxygen content – which initially decreases due to biological oxygen demand – also increases with depth. In contrast to the present-day scenario, we propose that during the Archaean and Proterozoic eons inverted profiles could have developed such that, with depth, ocean water temperature increased and density and dissolved oxygen decreased. These inverted temperature and density profiles resulted in palaeo-ocean circulation behaving as a free convective system. It is proposed that this free convection, which may have been stable, or chaotic and subject to secondary instabilities, hindered the deep oxygenation of the palaeo-ocean. It may not be coincidental that the great oxygenation event (GOE) and Huronian glaciations are contemporaneous, in a similar way that the Neoproterozoic oxygenation event (NOE) is known to have been associated with glaciations. The global-scale external forcing required to switch the natural convective system to its present-day configuration is suggested to have been associated with Neoproterozoic glaciations and the subsequent lowering of ocean water salinity that accompanied them. We propose that this inverted the ocean water density gradient, allowing the oxygenation of the oceans for the first time. It is beyond the scope of this work to model the complex natural convection system, but we hope that geophysicists and numerical modellers will quantitatively evaluate the hypothesis proposed here to validate or refute our proposition.

When a gas turbine engine is shut down it will develop a circumferential thermal gradient vertically across the compressor due to hot air rising from the cooling metal components and pooling at the top. As the hot compressor rotor drum and casing cool and contract in the presence of this thermal gradient, they do so non-uniformly and therefore will bend slightly, in a phenomenon known as rotor bow. Starting an engine under bowed conditions can result in damage, representing a risk to both airworthiness and operational capability. This study consolidates some preliminary findings by the authors relating to the drivers for rotor bow, such as engine geometry, aircraft-engine integration and rotor temperature on shutdown. The commercial and military operational considerations associated with rotor bow are also discussed, including limitations which may result from a bowed rotor; the influence of operations including the final flight and descent profiles, taxi procedures and rapid turnaround requirements; as well as some practical solutions which may be implemented to reduce the impact of rotor bow.

An investigation of natural convective flow and heat transfer inside a three dimensional rectangular cavity containing an array of discrete heat sources is carried out. The array consists of a row and columnwise regular arrangement of identical square shaped isoflux discrete heaters and is flush mounted on a vertical wall of the cavity. A symmetrical isothermal sink condition is maintained by cooling the cavity uniformly from either the opposite wall or the side walls or the top and bottom walls. The other walls of the cavity are maintained adiabatic. A finite volume method based on the SIMPLE algorithm and the power law scheme is used to solve the conservation equations. The parametric study covers the influence of pertinent parameters such as the Rayleigh number, the Prandtl number, side aspect ratio of the cavity and cavity heater ratio. A detailed fluid flow and heat transfer characteristics for the three cases are reported in terms of isothermal and velocity vector plots and Nusselt numbers. In general it is found that the overall heat transfer rate within the cavity for Ra=107 is maximum when the side aspect ratio of the cavity lies between 1.5 and 2. A more complex and peculiar flow pattern is observed in the presence of top and bottom cold walls which in turn introduces hot spots on the adiabatic walls. Their location and size are highly sensitive to the side aspect ratio of the cavity and hence offers more effective ways for passive heat removal.

In present work, we investigate numerical simulation of steady natural convection flow in the presence of weak magnetic Prandtl number and strong magnetic field by involving algebraic decay in mainstream velocity. Before passing to the numerical simulation, we formulate the set of boundary layer equations with the inclusion of the effects of algebraic decay velocity, aligned magnetic field and buoyant body force in the momentum equation. Later, finite difference method with primitive variable formulation is employed in the physical domain to compute the numerical solutions of the flow field. Graphical results for the velocity, temperature and transverse component of magnetic field as well as surface friction, rate of heat transfer and current density are presented and discussed. It is pertinent to mention that the simulation is performed for different values of algebraic decay parameter α, Prandtl number Pr, magnetic Prandtl number Pm and magnetic force parameter S.

Magnetohydrodynamic natural convection heat transfer in a rotating, differentially heated enclosure is studied numerically in this article. The governing equations are in velocity, pressure and temperature formulation and solved using the staggered grid arrangement together with MAC method. The governing parameters considered are the Hartmann number, 0≤Ha≤70, the inclination angle of the magnetic field, 0°≤θ 90°, the Taylor number, 8.9 x 104≤Ta≤1.1 x 106 and the centrifugal force is smaller than the Coriolis force and the both forces were kept below the buoyancy force. It is found that a sufficiently large Lorentz force neutralizes the effect of buoyancy, inertial and Coriolis forces. Horizontal or vertical direction of the magnetic field was most effective in reducing the global heat transfer.

The purely conductive state in configurations such as the Rayleigh–Bénard one is linearly stable for yield stress fluids at all Rayleigh numbers, $Ra$. However, on changing to localized heater configurations the static background state exists only if the yield stress is sufficiently large. Otherwise, thermal plumes may be induced in a stationary viscoplastic fluid layer, as illustrated in the recent experimental study of Davaille et al. (J. Non-Newtonian Fluid Mech., vol. 193, 2013, 144–153). Here, we study an analogous problem both analytically and computationally, from the perspective of an ideal yield stress fluid (Bingham fluid) that is initially stationary in a locally heated rectangular tank. We show that for a non-zero yield stress the onset of flow waits for a start time $t_{s}$ that increases with the dimensionless ratio of yield stress to buoyancy stress, denoted $B$. We provide a precise mathematical definition of $t_{s}$ and approximately evaluate this for different values of $B$, using both computational and semianalytical methods. For sufficiently large $B\geqslant B_{cr}$, the fluid is unable to yield. For the flow studied, $B_{cr}\approx 0.00307$. The critical value $B_{cr}$ and the start time $t_{s}$, for $B<B_{cr}$, are wholly independent of $Ra$ and $Pr$. For $B<B_{cr}$, yielding starts at $t=t_{s}$. The flow develops into either a weakly or a strongly convective flow. In the former case the passage to a steady state is relatively smooth and monotone, resulting eventually in a steady convective plume above the heater, rising and impinging on the upper wall, then recirculating steadily around the tank. With strongly convecting flows, for progressively larger $Ra$ we observe an increasing number of distinct plume heads and a tendency for plumes to develop as short-lived pulses. Over a certain range of $(Ra,B)$ the flow becomes temporarily frozen between two consecutive pulses. Such characteristics are distinctly reminiscent of the experimental work of Davaille et al. (J. Non-Newtonian Fluid Mech., vol. 193, 2013, 144–153). The yield stress plays a multifaceted role here as it affects plume temperature, size and velocity through different mechanisms. On the one hand, increasing $B$ tends to increase the maximum temperature of the plume heads. On the other hand, for larger $B\rightarrow B_{cr}$, the plume never starts.

A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presented in this paper for the simulation of both isothermal and thermal flows. The present solver combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) to solve the N-S equations. Different from the conventional N-S solvers, its viscous and inviscid fluxes at the cell interface are evaluated simultaneously by local reconstruction of LBE solution. As compared to the conventional LBE solvers, which apply the lattice Boltzmann method (LBM) globally in the whole computational domain, it only applies LBM locally at each cell interface, and flow variables at cell centers are given from the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS, the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-up of mesh spacing and time step, tedious implementation of boundary conditions, are completely removed. The accuracy, efficiency and stability of the proposed solver are examined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and natural convection. Numerical results show that the LBFS has a second order of accuracy in space. The efficiency of the LBFS is lower than LBM on the same grids. However, the LBFS needs very less non-uniform grids to get grid-independence results and its efficiency can be greatly improved and even much higher than LBM. In addition, the LBFS is more stable and robust.

A new solver is developed to numerically simulate the melting phase change with natural convection. This solver was implemented on a single Nvidia GPU based on the CUDA technology in order to simulate the melting phase change in a 2D rectangular enclosure. The Rayleigh number is of the order of magnitude of 108 and Prandlt is 50. The hybrid thermal lattice Boltzmann method (HTLBM) is employed to simulate the natural convection in the liquid phase, and the enthalpy formulation is used to simulate the phase change aspect. The model is validated by experimental data and published analytic results. The simulation results manifest a strong convection in the melted phase and a different flow pattern from the reference results with low Rayleigh number. In addition, the computational performance is estimated for single precision arithmetic, and this solver yields 703.31MLUPS and 61.89GB/s device to device data throughput on a Nvidia Tesla C2050 GPU.

We show that a single particle distribution for the “energy-conserving” D2Q13 lattice Boltzmann scheme can simulate coupled effects involving advection and diffusion of velocity and temperature. We consider various test cases: non-linear waves with periodic boundary conditions, a test case with buoyancy, propagation of transverse waves, Couette and Poiseuille flows. We test various boundary conditions and propose to mix bounce-back and anti-bounce-back numerical boundary conditions to take into account velocity and temperature Dirichlet conditions. We present also first results for the de Vahl Davis heated cavity. Our results are compared with the coupled D2Q9-D2Q5 lattice Boltzmann approach for the Boussinesq system and with an elementary finite differences solver for the compressible Navier-Stokes equations. Our main experimental result is the loss of symmetry in the de Vahl Davis cavity computed with the single D2Q13 lattice Boltzmann model without the Boussinesq hypothesis. This result is confirmed by a direct Navier Stokes simulation with finite differences.

This paper is a numerical work to investigate thermal interactions between heat fluxes generated by electronic devices mounted on a vertical printed circuit board (PCB). In fact, the effect of some parameters such as Grashof number (Gr), the distance (S) between the heat sources and the upper exit distance (Le) was studied. The results show that a regular and uniform heat sources distribution in the inlet is very important in order to take advantage from the leading edge of the boundary layer and to obtain the necessary heat dissipation. The impact of parameters on the heat dissipation, characterized by the Nusselt number, has a different importance. For a Prandtl number Pr = 0.71, the (Gr) increases the heat exchange that is reflected by the increase of Nusselt number, and also participates to the formation of recirculation zones. The total Nusselt number Nu is increased when (Gr) is multiplied by 102. With regards to the distance (S) between the heat sources, the results show that Nu increases about 7% if the distance (S) doubles, and becomes approximately 17.8% when (S) quadruples. At the end, the heat transfer increases when increasing the distance (Le) of the upper exit length, especially on the second component. The total Nusselt number increases by 1.6% when (Le) increases by about 67%.

In this paper, a boundary condition-enforced IBM is introduced into the LBM in order to satisfy the non-slip and temperature boundary conditions, and natural convections in a concentric isothermal annulus between a square outer cylinder and a circular inner cylinder are simulated. The obtained results show that the boundary condition-enforced method gives a better solution for the flow field and the complicated physics of the natural convections in the selected case is correctly captured. The calculated average Nusselt numbers agree well with the previous studies.

In this article, natural convection of a temperature-sensitive magnetic fluid in a porous media is studied numerically by using lattice Boltzmann method. Results show that the heat transfer decreases when the ball numbers increase. When the magnetic field is increased, the heat transfer is enhanced; however the average wall Nusselt number increases at small ball numbers but decreases at large ball numbers due to the induced flow being more likely confined near the bottom walls with a high number of obstacles.

We study the influence of natural convection on stability of reaction fronts in porous media. The model consists of the heat equation, of the equation for the depth of conversion and of the equations of motion under the Darcy law. Linear stability analysis of the problem is fulfilled, the stability boundary is found. Direct numerical simulations are performed and compared with the linear stability analysis.

Les transferts par convection naturelle sont fréquemment utilisés dans les différents procédés et sont également rencontrés dans diverses situations dans la nature. Afin d'améliorer ces transferts sans modifier la géométrie, il est possible de perturber l'écoulement d'origine par un obstacle poreux le long du trajet de l'écoulement principal. Cet obstacle modifie la structure de l'écoulement et affecte les transferts locaux. Cette modification est plus prononcée dans une configuration cylindrique au vu de la dissymétrie des écoulements résultants. La variation temporelle du transfert de chaleur est analysée pour différentes courbures et différentes valeurs du nombre de Darcy dans le milieu poreux. Cela nous permettra d'étudier les trois configurations : l'absence de l'obstacle (cas fluide), sa présence (bloc solide) et le cas intermédiaire du bloc poreux. Les conditions nécessaires pour améliorer le transfert dans chaque configuration sont illustrées.

This work studies the natural convection heat and mass transfer near a horizontal cylinder of elliptic cross section with constant wall temperature and concentration in a fluid-saturated porous medium. A coordinate transformation is used to obtain the nonsimilar governing boundary layer equations. The transformed governing equations are then solved by the cubic spline collocation method. Results for the local Nusselt and Sherwood numbers are presented as functions of the Lewis number, the buoyancy ratio, and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). The heat and mass transfer rates of the elliptical cylinder with slender orientation are higher than those with blunt orientation.

In the second part of the paper, we compare the solutions producedin the framework of the conference “Mathematical and numericalaspects of low Mach number flows” organized by INRIA and MAB inPorquerolles, June 2004, to the reference solutions described inPart 1. We make some recommendations on how to produce goodquality solutions, and list a number of pitfalls to be avoided.

There are very few reference solutions in the literature onnon-Boussinesq natural convection flows. We propose here a testcase problem which extends the well-known De Vahl Davisdifferentially heated square cavity problem to the case of largetemperature differences for which the Boussinesq approximation isno longer valid. The paper is split in two parts: in this firstpart, we propose as yet unpublished reference solutions for casescharacterized by a non-dimensional temperature difference of 0.6,Ra = 106 (constant property and variable property cases) andRa = 107 (variable property case). These reference solutions wereproduced after a first international workshop organized by CEA andLIMSI in January 2000, in which the above authors volunteered toproduce accurate numerical solutions from which the presentreference solutions could be established.