Longitudinal Convection Rolls Research Articles

Linear stability of longitudinal convective rolls in a non-Darcy porous layer filled with nanofluid due to viscous dissipation effect: A realistic approach

The onset of longitudinal convective rolls of nanofluid flow inside a horizontal porous material using Brinkman theory due to the viscous dissipation effect has been addressed numerically. An infinitely long horizontal porous region has been considered which is confined between two rigid boundaries. The boundaries of the channel are considered adiabatic (lower) and isothermal (upper). Brownian motion and thermophoresis properties are taken into consideration for nanofluid. In the linear stability analysis, oblique roll disturbances with arbitrary orientation are taken into account, where the preferred mode of instability i.e., longitudinal rolls (when P=0 or ϑ=π/2) are shown. Finally, the resulting eigenvalue problem is solved in MATLAB using the bvp4c technique. The impact of improving the modified diffusivity ratio (NA<10), Lewis number (Le<,=,>1), concentration Rayleigh number (Rn) and modified particle density increment (NB≈10−2−10−5) is to accelerate the instability boundaries, while the switching parameter (ξ) has a dual character on it. Further, the pattern of the streamlines are slightly compressed in the channel’s center for large values of ξ, whereas no markable changes are noticed for higher values of NA. The isotherms lines are closer near the adiabatic boundary when ξ takes small values.

THE EFFECT OF ROTATION AND PULSATING THROUGHFLOW ON THE ONSET OF LONGITUDINAL CONVECTIVE ROLLS IN A POROUS MEDIUM SATURATED BY NANOFLUID

This study accentuates the combined effect of rotation and pulsating throughflow on the onset of longitudinal convec-tive rolls in a porous medium saturated by nanofluid. Pulsating throughflow modifies the basic profiles for temperature and the volumetric fraction of nanoparticle from linear to nonlinear with layer height, which in turn affects the stability significantly. To treat this problem, Buongiorno's two-phase mathematical model is used. Using the framework of linear stability theory and the frozen profile approach, the stability equations are derived and solved analytically using the Galerkin weighted residuals method. The effect of increasing the modified diffusive ratio NA and the nanoparticle Rayleigh-Darcy number RN, and decreasing the rotation parameter TD is to accelerate the onset of convection, while the Lewis number Le and porosity parameter φ have dual effect on the stability of the system in the presence of pulsating throughflow. Moreover, the frozen profile approach shows that the effect of pulsating throughflow in both directions is stabilizing. An increased amplitude of throughflow oscillations leads to increased stability by an amount that depends on frequency.

Linear instability analysis of the onset of thermal convection in an Ekman-Couette-flow

The onset of thermal convection in a plane layer with rotation and shear is investigated. The boundaries of the layer are parallel to the x−y plane of the Cartesian coordinate system. The layer rotates at a constant angular velocity Ωz around the z axis, and is sheared by moving, along the x direction, the lower and upper boundaries parallel to themselves with constant velocity −Uw and Uw respectively. The temperature of the lower boundary is higher than the temperature of the upper boundary. The linear instability equations are solved by using the Tau-Chebyshev spectral numerical method. The critical parameters of the stationary, transverse and longitudinal convective rolls are presented. The relationship between the critical Rayleigh number and: (i) the Taylor number, (ii) the wavenumber vector, (iii) the Reynolds number, (iv) the kinetic energy norm and (v) the heat transfer at the sliding plates, is presented. We find the combination of the critical parameters that lead to the occurrence of two interesting situations, these are (a) a high value of the kinetic energy norm which promotes the formation of inertial longitudinal rolls, which are considered as hydrodynamic instabilities modified by the buoyancy force, and (b) a small value of the kinetic energy norm together with a high value of the heat transfer at the sliding plates. The latter situation, may be considered in the single-crystal growth from the melt processes.

The onset of longitudinal convective rolls in a porous medium saturated by a nanofluid with non-uniform internal heating and chemical reaction

The aim of this article is to investigate the combined effect of chemical reaction and non-uniform internal heating on the onset of longitudinal convective rolls in a porous medium saturated by nanofluid. Non-uniform internal heating modifies the basic profiles for temperature and concentration of nanoparticle from linear to nonlinear with layer height, which in turn affects the stability extensively. To treat this problem, the Buongiorno’s two-phase mathematical model is used for nanofluid, while for porous medium Darcy model is utilized. Using the framework of linear stability theory, the stability equations are derived and solved analytically using the Galerkin weighted residuals method. Results show that the effect of increasing the Lewis number $$L_{\text{e}}$$ , internal heating parameter $$S$$ , modified diffusive ratio $$N_{\text{A}}$$ and the nanoparticle Rayleigh–Darcy number $$R_{\text{N}}$$ is to accelerate the onset of convection, while the chemical parameter $$K_{\text{r}}$$ and porosity parameter $$\varepsilon$$ is to delay the onset of convection. The size of convection cell increases with the increase in internal heating parameter $$S$$ and porosity parameter $$\varepsilon$$ , while it decreases with the chemical parameter $$K_{\text{r}}$$ , the Lewis number $$L_{\text{e}}$$ , modified diffusive ratio $$N_{\text{A}}$$ and the nanoparticle Rayleigh–Darcy number $$R_{\text{N}}$$ .

Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field

The effect of a uniform external magnetic field on the onset of convection in an electrically conducting nanofluid layer is examined numerically based on non-homogeneous two-phase model (i.e., classical Buongiorno’s mathematical model) which incorporates the effects of Brownian motion and thermophoresis of nanoparticles in the thermal transport mechanism of nanofluids. In this investigation, we consider that the nanofluid is Newtonian, heated from below and confined horizontally in a Darcy-Brinkman porous medium between two infinite rigid boundaries, with different nanoparticle configurations at the horizontal boundaries (i.e., top heavy and bottom heavy nanoparticle distributions). The linear stability theory has been wisely used to obtain a set of linear differential equations which are transformed to an eigenvalue problem, so that the thermal Rayleigh number Ra is the corresponding eigenvalue. The thermal Rayleigh number Ra and its corresponding wave number a are found numerically using the Chebyshev-Gauss-Lobatto collocation method for each set of fixed nanofluid parameters. The marginal instability threshold (Rac, ac) characterizing the onset of stationary convection is computed accurately for wide ranges of the modified magnetic Chandrasekhar number Q, the modified specific heat increment NB, the nanoparticle Rayleigh number RN, the modified Lewis number Le, the modified diffusivity ratio NA and the Darcy number Da. Based on these control parameters and the notions of streamlines, isotherms and iso-nanoconcentrations, the stability characteristics of the system and the development of complex dynamics at the critical state are discussed in detail for both nanoparticle distributions.

CFD analysis for solar chimney power plants

Solar chimney power plants are investigated numerically using ANSYS Fluent and an in-house developed Computational Fluid Dynamics (CFD) code. Analytical scaling laws are verified by considering a large range of scales with tower heights between 1m (sub-scale laboratory model) and 1000m (largest envisioned plant). A model with approximately 6m tower height is currently under construction at the University of Arizona. Detailed time-dependent high-resolution simulations of the flow in the collector and chimney of the model provide detailed insight into the fluid dynamics and heat transfer mechanisms. Both transversal and longitudinal convection rolls are identified in the collector, indicating the presence of a Rayleigh–Bénard–Poiseuille instability. Local separation is observed near the chimney inflow. The flow inside the chimney is fully turbulent.

A numerical study of the longitudinal thermoconvective rolls in a mixed convection flow in a horizontal channel with a free surface

This paper presents a numerical study of three-dimensional laminar mixed convection within a liquid flowing on a horizontal channel heated uniformly from below. The upper surface is free and assumed to be flat. The coupled Navier–Stokes and energy equations are solved numerically by the finite volume method taking into account the thermocapillary effects (Marangoni effect). When the strength of the buoyancy, thermocapillary effects and forced convective currents are comparable (Ri⋍O(1) and Bd=Ra/Ma⋍O(1)), the results show that the development of instabilities in the form of steady longitudinal convective rolls is similar to those encountered in the Poiseuille–Rayleigh–Bénard flow. The number and spatial distribution of these rolls along the channel depend on the flow conditions. The objective of this work is to study the influence of parameters, such as the Reynolds, Rayleigh and Biot numbers, on the flow patterns and heat transfer characteristics. The effects of variations in the surface tension with temperature gradients (Marangoni effect) are also considered.

Selection of convective rolls in a thin evaporating-fluid layer in an air flow

The process of selection of longitudinal convective rolls in a thin layer of evaporating fluid immersed in an air turbulent boundary layer flow is studied numerically. The dependence of the two-dimensional flow patterns on the Rayleigh number and boundary conditions is analyzed. Calculations with account for the thermocapillary effect are carried out. The numerical results are compared with experimental data.

Suppressing falling film instabilities by Marangoni forces

The linear stability of a thin liquid layer falling down an inclined wall heated by a downstream linearly increasing temperature distribution is investigated. It is shown that hydrodynamic and Marangoni instabilities yield two types of transverse instabilities: long surface waves and convective rolls, and longitudinal convective rolls, much like in the case of a uniformly heated wall. However, in contrast to the problem of a uniformly heated wall, where the thermocapillary forces have a destabilizing influence on all instability modes, here they can either destabilize or stabilize the flow. For liquids with sufficiently large Prandtl numbers, increasing the temperature gradient first destabilizes the flow and then stabilizes it. On the other hand, for small Prandtl numbers, increasing the temperature gradient leads to a monotonic stabilization of all instability modes.

On Magnetic Field Control Experiments of Ferrofluid Convection Motion

Summary The experiments have been performed on the stability of buoyancy-driven flows of a ferrofluid for the cases of an inclined and vertical orientation of thin cylindrical encloser. Visualization of flow patterns is provided by a temperature-sensitive liquid crystal sheet which serves as the lower surface of a transparent heat exchanger. The local temperature sensor is used for measurement of heat transport across the fluid layer. The results indicate that with the help of an external homogeneous transversal magnetic field it is possible to control the stability of thermogravitational shear flow under vertical orientation of the layer, i.e. induce the thermomagnetic mode of instability with vertical orientation of longitudinal convection rolls. As numerous experimental investigations show the origin of concentration density gradients due to gravity sedimentation of magnetic particles and their aggregates results in traveling wave regimes of Rayleigh and thermomagnetic convection flows.

Symmetry of flow in the Comstock Lode hydrothermal system: Evidence for longitudinal convective rolls in geologic systems

Hydrothermal fluid flow can be visualized by three‐dimensional (3‐D) kriging and computer contouring of oxygen isotope data on propylitized andesites in the Comstock Lode mining district. This data set consists of δ18O results for 327 samples from surface outcrops, drill cores, and mine workings in a 3.5 km × 1.7 km × 1.0 km block in the hanging wall of the Comstock fault encompassing the region where the most important ore bodies were discovered. The δ18O contours deep in the hanging wall are not simply parallel to the Comstock fault or to the boundary of the Davidson granodiorite stock as would be expected for a hydrothermal convection system dominated by unicellular flow. Instead, the subsurface δ18O values exhibit a high degree of symmetry about a vertical plane directed radially toward the stock, with isotopic contours being predominantly perpendicular to the Comstock fault and the eastern contact of the Davidson granodiorite. A vertical cross section along the Comstock Lode shows a low δ18O core with two symmetrical zones of higher δ18O values to either side where the contours are steeply inclined, which we infer reflects downwelling of fluids shifted by water‐rock exchange or steam loss. This 3‐D visualization of δ18O values provides the first evidence that longitudinal convective rolls can be significant in geologic environments, in this case within a zone of high temperature gradients and permeability near the Comstock fault and the Davidson granodiorite stock.

Physics of multiscale convection in Earth's mantle: Evolution of sublithospheric convection

We investigate the physics of multiscale convection in Earth's mantle, characterized by the coexistence of large‐scale mantle circulation associated with plate tectonics and small‐scale sublithospheric convection. In part 2 of our study, the temporal and spatial evolution of sublithospheric convection is studied using two‐dimensional whole mantle convection models with temperature‐ and depth‐dependent viscosity and an endothermic phase transition. Scaling laws for the breakdown of layered convection as well as the strength of convection are derived as a function of viscosity layering, the phase buoyancy parameter, and the thermal Rayleigh number. Our results suggest that layered convection in the upper mantle is maintained only for a couple of overturns, with plausible mantle values. Furthermore, scaling laws for the onset of convection, the stable Richter rolls, and the breakdown of layered convection are all combined to delineate possible dynamic regimes beneath evolving lithosphere. Beneath long‐lived plates, the development of longitudinal convection rolls is suggested to be likely in the upper mantle, as well as its subsequent breakdown to whole mantle‐scale convection. This evolutionary path is suggested to be consistent with the seismic structure of the Pacific upper mantle.

Secondary flow and enhancement of heat transfer in horizontal parallel-plate and convergent channels heating from below

Experimental studies of secondary air flow structure and enhancement of heat transfer in horizontal parallel-plate and convergent channels have been carried out. The bottom wall is horizontal and heated uniformly, while the opposite wall is insulated and inclined with respect to the horizontal plate so as to create a convergence angel of 3° for the convergent channel. The aspect ratio (width to height) and the ratio of channel length to height at the entrance of the channel is 6.67 and 15, respectively. The Reynolds number ranges from 100 to 2000, the buoyancy parameter, Gr/ Re 2, from 2.5 to 907 and Pr of the air flow is 0.7. Flow structure inside the channel is visualized by injecting smoke at the inlet flowing along the bottom wall. The complete processes for the formation of the plumes associated with vortices and their transformation into longitudinal convection rolls due to the lateral extension and combination of the vortices are observed. The number of convection rolls formed is much less than those found in the experiments with water. For the convergent channel,the favorable pressure gradient causes a thinner bottom heated layer which results in much later initiation of secondary flow and fewer the number of plume produced. The interactions between neighboring vortices and plumes are suppressed by the acceleration of mainstream, and results ina stable flow and less pronounced enhancement of heat transfer. Temperature fluctuations at different locations are measured to indicate the flow structure and oscillation of the secondary flow. The effects of the buoyancy parameter and the convergence of the channel on the secondary flow structure and the Nusselt number are presented and discussed.

Transport By Gravity Currents In Building Fires

Gravity currents (GC) are important physical phenomena which transport smoke and hot gases id corridors of buildings. In this paper, they are studied using l q e eddy simulations (LES). The transient Navier-Stokes (N-S) equations are numerically integrated with very high resolution, in two dimensions and with high resolution in three dimensions. The LES computations require no adjustable parameters, and are found to agree well with available experimental results in the absence of heat transfer. Fresh-water/salt-water experimental results are compared with computational results with adiabatic boundary conditions to demonstrate the validity of the model. Whereas, a high-resolution two-dimensional computation simulates very well the gravity current with adiabatic wall boundary conditions, significant wall heat transfer requires the consideration of three-dimensional effects. Heat transfer to the ceiling of a corridor from an underlying GC produces longitudinal convection rolls which significantly enhance heat losses to the ceiling and reduce the speed of progression of the GC down the corridor.

Numerical experiments on laminar natural convection in rectangular cavities with and without honeycomb‐structures

Two‐dimensional finite difference calculations are carried out to study laminar flow in longitudinal and transverse convection rolls for three different geometries: a single rectangular cavity with high aspect ratio; a double cavity with a thin separation sheet; and a double cavity with a separation sheet and a honeycomb structure. The equations for the convection‐diffusion in the fluid and conduction in the solid region are solved simultaneously. Good agreement with experimental data is achieved for Rayleigh numbers not too high above the critical value for the onset of secondary convection rolls (Ra &lt; 8500 for vertical and Ra &lt; 2700 for horizontal cavities filled with air). Simulation fails for inclined cavities, where the flow structure is essentially three‐dimensional.

Interaction between longitudinal convection rolls and transverse waves in unstably stratified plane Poiseuille flow

Nonlinear interaction between a Tollmien–Schlichting wave and longitudinal rolls resulting from Rayleigh–Bénard instability has been investigated in an unstably stratified plane Poiseuille flow of infinite extent. Cubic order amplitude equations for the interacting modes are derived on a weakly nonlinear basis in the neighborhood of a crossover point at which both modes become unstable on a linear basis simultaneously. A bifurcation analysis based on use of the actual numerical coefficients obtained from the governing equations and evaluated for various values of the Prandtl number is performed, and the results, such as the effect of longitudinal rolls on the subcritical instability characteristic of a Tollmien–Schlichting wave, are discussed.

Rayleigh-Bénard problem with imposed weak through-flow: Two coupled Ginzburg-Landau equations.

The Rayleigh-B\'enard instability of a horizontal shear flow in a narrow channel is governed by two competing unstable modes: traveling transverse and stationary longitudinal convection rolls. The dynamics of the mode amplitudes obeys two coupled Ginzburg-Landau equations, which we derive in the present article. For mathematical simplicity we assume free-slip boundaries at the channel sidewalls. The analytical discussion and the numerical simulation of these amplitude equations show transitions between the two convective structures. Propagating fronts, which separate areas of transverse rolls and longitudinal ones, occur in the parameter range where both patterns are stable. Coexisting uniform states with nonzero amplitudes of both modes are unstable solutions of the equations. Results are in qualitative agreement with recent experiments.

Simple model for the Bénard instability with horizontal flow near threshold.

We present a simple model for Rayleigh-B\'enard convection with an imposed horizontal flow of Reynolds number R in a container of finite width and near threshold. The model consists of two coupled envelope equations representing longitudinal and transverse traveling convection rolls. Each equation is for a wave traveling in the direction of the imposed flow. For small R, transverse rolls are stable at the convective onset. For R&gt;${\mathit{R}}^{\mathrm{*}}$, longitudinal rolls bifurcate from the conduction state. For a range of cross-coupling strengths and for R&gt;${\mathit{R}}^{\mathrm{*}}$, we obtain a transition from longitudinal to transverse rolls as the Rayleigh number is increased. This transition occurs via states for which part of the system is occupied by longitudinal, and another by transverse rolls. The behavior is strongly influenced by the presence of noise since the system first becomes convectively unstable, and therefore noise-sustained structures can play an important role. We also show that for a range of parameters in the model, a mixed state (for which both envelopes assume a nonzero value at the same location) is possible over part of the cell in a finite geometry.

Three-dimensional thermal cellular convection in rectangular boxes

The extension of the classic Rayleigh–Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of inter-ferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects.

Mixed convection between horizontal plates—II. Fully developed flow

Fully developed velocity profiles of longitudinal convection rolls in mixed convection between horizontal plates were measured in nitrogen by laser Doppler anemometry for a range 2472 < Ra < 8300 and 15 < Re < 150. It is shown analytically and experimentally that the transverse velocities of the longitudinal convection rolls are independent of the forced flow. The experimentally and numerically obtained w-profiles ( Pr = 0.71) are in good agreement with theoretical predictions ( Pr → ∞) and other experimental results ( Pr = 11.1 and 930) for Rayleigh-Benard convection. A detailed study of the longitudinal velocity modulation Δu[ w max ( Ra), Re] is presented. Also, asymmetric roll patterns were found in spite of the small temperature differences used between the horizontal plates.