The nonlinear development of finite amplitude disturbances in mixed convection flow in a heated vertical annulus is studied by direct numerical simulation. The unsteady Navier Stokes equations are solved numerically by a spectral method for different initial conditions. The results indicate that the equilibrium state of the flow is not unique, but depends on the amplitude and wavenumber of the initial disturbance. In all cases, the equilibrium state consists of a single dominant mode with the wavenumber kf, and its superharmonics. The range of equilibrium wavenumbers kf was found to be narrower than the span of the neutral curve from linear theory. Flows with wavenumbers outside this range, but within the unstable region of linear theory are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that linearly stable long and short waves can also excite a wave inside this narrow band through nonlinear wave interaction. The results suggest that the selection of the equilibrium wavenumber kf is due to a nonlinear energy transfer process which is sensitive to initial conditions.
Ortiz-Pérez and Dávalos investigated the stability of parallel convective flow. From the article it seems that this equations of flow obtained by the authors. Meanwhile, it is well studied theoretically and experimentally(exact solution obtained more than half a century ago.
Comment on Conjugate heat transfer of mixed convection for viscoelastic fluid past a stretching sheet, by Hsiao and Chen, Mathematical Problems in Engineering
I consider a dielectric fluid heated from above and subjected to an electric potential difference between its top and bottom. I show that for a suitably chosen electric potential difference, the layer of fluid can become unstable. For the case of a strongly polar fluid like pure water, an electric potential difference of a few hundreds of volts can trigger convection. Although the analysis in this paper cannot explain the phenomenon described in [Gross & Porter(1966)], it could be because of unavailability of accurate physical parameters of the fluid used in the experiment.
We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection i…
Gravity-driven convection of fluid at parameters near its thermodynamic critical point inside a porous layer heated from below (Rayleigh-Darcy convection) is studied. The fluid having the temperature slightly above the critical one is one-component. The hydrodynamic model describing a high compressible fluid phase at variable physical properties inside a solid matrix at a uniform porosity is analyzed. A near-critical fluid is assumed to be the van der Waals gas. In the limit of small variations in the density and thermodynamic coefficients where the Oberbeck-Boussinesq approximation is applica…
A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches -- linear stability analysis, fully nonlinear numerical simulations, and a coarse-grained model -- to determine the circumstances that lead to each configuration. The coarse-grained model yields…
Scaled physical modeling is an important means to understand the behavior of fluids in nature. However, a common source of errors is conflicting similarity criteria. Here, we present using hypergravity to improve the scaling similarity of gravity-dominated fluid convection, e.g. natural convection and multi-phase flow. We demonstrate the validity of the approach by investigating water-brine buoyant jet experiments conducted under hypergravity created by a centrifuge. Results show that the scaling similarity increases with the gravitational acceleration. In particular, the model best represents…
Multi-fluid models have recently been proposed as an approach to improving the representation of convection in weather and climate models. This is an attractive framework as it is fundamentally dynamical, removing some of the assumptions of mass-flux convection schemes which are invalid at current model resolutions. However, it is still not understood how best to close the multi-fluid equations for atmospheric convection. In this paper we develop a simple two-fluid, single-column model with one rising and one falling fluid. No further modelling of sub-filter variability is included. We then ap…
We perform linear and nonlinear stability analysis for thermal convection in a fluid overlying a saturated porous medium. We use a coupled system, with the Navier-Stokes equations and Darcy's equation governing the free-flow and the porous regions respectively. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid-medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. Within certain regimes, the nonlinear stability thresholds agree cl…
A transition to coordinates formed by streamlines and orthogonal ones are convenient to simplify the Navier-Stokes and Fourier-Kirchoff system. We derive transformation to such coordinates, taking into account a necessity to introduce integrating factor that is not equal to unity in a viscous flow. The transition allows to express approximately velocity module and the velocity vector inclination to vertical direction in terms of temperature gradient in explicit form.
This investigation deals with the analysis of stagnation point heat transfer and corresponding flow features of hydromagnetic viscous incompressible fluid over a vertical shrinking sheet. The considered sheet is assumed to be permeable and subject to addition of stagnation point to control the generated vorticity in the boundary layer. The sheet is placed on the right side of the fluid saturated porous medium which is having permeability of specified form. Nonlinear convection waves in the flow field are realized due to the envisaged nonlinear relation between density and temperature. The equa…
In Rayleigh-Bénard convection, it has been found that the amount of heat passing through the fluid has a power law dependence on the imposed temperature difference. Modifying this dependence, either enhancing or reducing the heat transfer capability of fluids, is important in many scientific and practical applications. Here, we present a simple means to control the vertical heat transfer in Rayleigh-Bénard convection by injecting heat through one lateral side of the fluid domain and extracting the same amount of heat from the opposite side. This horizontal heat flux regulates the large-scale c…
In the above paper the authors treat the boundary layer flow along a stationary, vertical, permeable, flat plate within a vertical free stream. Fluid is sucked or injected through the vertical plate. The fluid species concentration at the plate is constant and different from that of the ambient fluid. It is also assumed that the plate is heated by convection from another fluid with constant temperature with a constant heat transfer coefficient. The temperature and species concentration difference between the plate and the ambient fluid creates buoyancy forces and the flow is characterized as m…
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh-Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh--Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation …
Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows prop…
Rayleigh-Bénard convection, i.e. the flow of a fluid between two parallel plates that is driven by a temperature gradient, is an idealised setup to study thermal convection. Of special interest are the statistics of the turbulent temperature field, which we are investigating and comparing for three different geometries, namely convection with periodic horizontal boundary conditions in three and two dimensions as well as convection in a cylindrical vessel, in order to work out similarities and differences. To this end, we derive an exact evolution equation for the temperature probability densit…
We consider a vertical cavity composed of two chambers separated by a retractable thermally insulated thin membrane. The upper and lower chambers are filled with an incompressible Boussinesq fluid and maintained at temperatures $T_2$ and $T_1>T_2$, respectively by two separate heaters. Upon retraction of the membrane, the two fluid masses form an unstably stratified configuration with cold and heavy fluid overlying a warmer and lighter fluid and separated by a non-free interface across which there is a jump in the density. The aim of this paper is to determine the threshold conditions for conv…
The chapter begins with a description of the annulus model of rotationg convection. The basic equations for the spherical problem are then introduced and the onset of columnar convection in spherical shells is discussed. The onset of inertial mode convection is described which prevails at very low Prandtl numbers. The properties of finite amplitude convection are outlined for a wide range of Prandtl numbers. Equatorially attached convection is considered which evolves from inertial mode convection.
Laminar natural convection from two horizontally aligned isothermal cylinders in unconfined Power-law fluids has been investigated numerically. The effect of horizontal spacing (0<=(S/D)<=10) on both momentum and heat transfer characteristics has been delineated under the following pertinent parameters: Grashof number (10<=Gr<=1e3), Prandtl number (0.71<=n<=100), and Power-law index (0.4<=n<=1.6). The heat transfer characteristics are elucidated in terms of isotherms, local Nusselt number (Nu) distributions and average Nusselt number values, whereas the flow characteristics are interpreted in …
When two fluids of different electrical conductivities are transported side by side in a microfluidic device under the influence of an electric field, an electrokinetic instability (EKI) is often generated after some critical values of the applied electric field strength and conductivity ratio. Many prior experimental and numerical studies show that this phenomenon results in a chaotic flow field inside a microdevice, thereby facilitating the mixing of two fluids if they are Newtonian in behaviour. However, the present numerical study shows that this chaotic convection arising due to the elect…
We report a numerical study of Rayleigh--Bénard convection through random porous media using pore-scale modelling, focusing on the Lagrangian dynamics of fluid particles and heat transfer for varied porosities $ϕ$. Due to the interaction between the porous medium and the coherent flow structures, the flow is found to be highly heterogeneous, consisting of convection channels with strong flow strength and stagnant regions with low velocities. The modifications of flow field due to porous structure have a significant influence on the dynamics of fluid particles. Evaluation of the particle displa…
Buoyancy effect on low-flow condition convective heat transfer of non-conventional coolants, such as liquid metal and molten salts, is a crucial safety factor to advanced reactors under transient or accidental scenarios. The distinct heat transfer characteristics of non-unitary Prandtl fluids and the inherent complexity of the low-flow mixed convection phenomena requires the development of novel turbulent and heat transfer models that are adaptive to different spatiotemporal scales involved in the mixed convection heat transfer. In this work, direct numerical simulation of low-flow mixed conve…
The present work investigates the internal flow structure of two condensing droplets of aqueous solution during interaction. The velocity of fluid inside the droplets was investigated by confocal micro-PIV technique. Condensation on the droplets was carried out inside a closed chamber by creating a difference in vapor pressure between the droplet interface and a reservoir fluid surrounding the droplet in the room temperature without any cooling. Condensation on the droplet leads to spatial variation of solute concentration inside the droplet causing buoyancy driven Rayleigh convection. Fluid f…
This report summarises the results for the numerical simulation of thermocapillary convection in a two-fluid system with a deformable interface. An explicit technique with 3rd order Runge-Kutta method in time, 2nd order ENO for the advection terms, and 2nd order central-differencing for the diffusion terms are employed in the momentum equations for simulating the flow on a staggered grid using a Marker and Cell method. An energy equation is solved numerically and a level-set method is used to implicitly capture the interface. A constant contact angle condition is assumed between the end walls …
A fluid dynamics video is linked to this article, which have been submitted to the Gallery of Fluid Motion as part of the 65th American Physical Society meeting of the Division of Fluid Dynamics, held in San Diego, California, USA, over 17-20 November 2012. The video serves to visualize flows generated in a rectangular enclosure that are subjected to both mechanical and thermal forcing through a common horizontal boundary. This system exhibits features consistent with either horizontal convection or lid-driven cavity flows depending on the ratio between thermal and mechanical stirring, and thr…
Thermal convection of fluid is a more efficient way than diffusion to carry heat from hot sources to cold places. Here, we experimentally study the Rayleigh-Bénard convection of aqueous glycerol solution in a cubic cell with suspensions of rod-like particles made of polydimethylsiloxane (PDMS). The particles are inertial due to their large thermal expansion coefficient and finite sizes. The thermal expansion coefficient of the particles is three times larger than that of the background fluid. This contrast makes the suspended particles lighter than the local fluid in hot regions and heavier in…
We discuss the thermal conduction and convection of thermally relativistic fluids between two parallel walls under the gravitational force, both theoretically and numerically. In the theoretical discussion, we assume that the Lorentz contraction is ignored and spacetime is flat. For understanding of the thermal conduction and convection of thermally relativistic fluids between two parallel walls under the gravitational force, we solve the relativistic Boltzmann equation using the direct simulation Monte Carlo method. Numerical results indicate that strongly nonequilibrium states are formed in …
Self-rewetting fluids (SRFs), such as aqueous solutions of long-chain alcohols, exhibit anomalous quadratic dependence of surface tension on temperature having a minimum and with a positive gradient. When compared to the normal fluids (NFs), the SRFs can be associated with significantly modified interfacial dynamics, which have recently been exploited to enhance flow and thermal transport in various applications (e.g., microgravity and microscale transport systems). In this work, first, we develop a new analytical solution of thermocapillary convection in superimposed two SRF layers confined w…
Thermal convection in fluid layers heated from below are usually realized experimentally as well as treated theoretically with fixed boundaries on which conditions for the temperature and the velocity field are prescribed. The thermal and velocity boundary layers attached to the upper and lower boundaries determine to a large extent the properties of turbulent convection at high Rayleigh numbers. Fixed boundaries are often absent in natural realizations of thermal convection. This paper studies the properties of convection driven by a planar heat source below a cooling source of equal size imm…
Understanding large scale circulations (LSCs) in turbulent convective systems is important for the study of stars, planets and in many industrial applications. The canonical model of the LSC is quasi-planar with additional horizontal sloshing and torsional modes [Brown E, Ahlers G (2009) J. Fluid Mech. 638:383--400; Funfschilling D, Ahlers G (2004) Phys. Rev. Lett. 92(19):194502; Xi HD et al. (2009) Phys. Rev. Lett. 102(4):044503; Zhou Q et al. (2009) J. Fluid Mech. 630:367--390]. Using liquid gallium as the working fluid, we show via coupled laboratory-numerical experiments that the LSC in a …
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
We numerically investigate turbulent Rayleigh-Bénard convection within two immiscible fluid layers, aiming to understand how the layer thickness and fluid properties affect the heat transfer (characterized by the Nusselt number $Nu$) in two-layer systems. Both two- and three-dimensional simulations are performed at fixed global Rayleigh number $Ra=10^8$, Prandtl number $Pr=4.38$, and Weber number $We=5$. We vary the relative thickness of the upper layer between $0.01 \le α\le 0.99$ and the thermal conductivity coefficient ratio of the two liquids between $0.1 \le λ_k \le 10$. Two flow regimes …
We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the …
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number ($Pr = 0.1$) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box ($L_x \times L_y \times 1$) and a wide range of Taylor numbers ($750 \le Ta \le 5000$) for the purpose. The horizontal aspect ratio $η= L_y/L_x$ of the box was varied from $0.5$ to $10$. The primary instability appeared in the …
The present work focuses on the study of mixed convection of a purely viscous shear-thinning fluid in a horizontal annular eccentric duct. The inner and outer cylinders are heated with constant and uniform heat flux densities. The objective of this work is to study the effect of the variation of eccentricity, rheological behavior of the fluid as well as the thermodependency of the rheological parameters on the reorganization of the flow and thermal stratification caused by the buoyancy forces. At the entrance of the heating zone, the dynamic regime is assumed to be established and the temperat…
The report elaborates for the first time visual observations of induced convections in a non-conducting diamagnetic fluid under the action of static gradient magnetic field in the absence of thermal gradients and the techniques employed to observe and record them. Suspension of Deionized (DI) and double distilled water and Lycopodium pollen grains was used as the fluid in a test tube. Permanent magnets having field strength of 0.12T each were used to provide the static gradient magnetic field. The suspension filled test tube was kept in a room temperature water bath. The convections were visua…
Effect of interfacial disturbances on instabilities of buoyant/thermocapillary convective flows in rectangular cavities is studied in a series of numerical experiments. The computations are carried out for several two-liquid two-layer systems taking into account properties of liquids used in previously published experiments. Relation between the interface deformations and the Boussinesq approximation is discussed. It is shown that in some systems, including the interface disturbances in the model can alter the critical temperature difference by approximately 10%, producing either destabilizing…
We consider the onset of Boussinesq convection in a horizontal layer of electrically conducting incompressible fluid with rigid electrically insulating horizontal boundaries. The fluid is heated from below and rotates about a vertical axis; vertical magnetic field is imposed. The instability of the trivial steady state (fluid at rest) can be monotonic or oscillatory depending on parameters of the problem (the kinematic and magnetic Prandtl, Taylor and Chandrasekhar numbers). If the Taylor number is sufficiently large, convective rolls, emerging in the monotonic instability, experience the Küpp…
It is well accepted that an outward Marangoni convection from a low surface tension region will make the surface depressed. Here, we report that this established perception is only valid for thin liquid films. Using surface laser heating, we show that in deep liquids a laser beam actually pulls up the fluid above the free surface generating fountains with different shapes. Whereas with decreasing liquid depth a transition from fountain to indentation with fountain in-indentation is observed. Further, high-speed imaging reveals a transient surface process before steady elevation is formed, and …
