Evolution to chaotic mixed convection in a multiple ventilated cavity

The characteristics of transition from laminar to chaotic mixed convection in a two-dimensional multiple ventilated cavity is analyzed in this paper. The horizontal air streams enter the cavity from the two inflow-openings near the top of both vertical walls, while the outflow openings are near the bottoms of both vertical walls. The results obtained for a range of the Richardson number, Ri, from 0.01 to 5 at Pr = 0.71, the Reynolds number, Re, from 1000 to 2500 and the inlet flow angle, φ, based on 0°, 20°, 45° and 70°. The results show that, as Ri increases, the solution may exhibit a change from steady-state to periodic oscillation, and then to non-periodic oscillatory state. However, the flow inside the cavity becomes steady-state again as Richardson number increases further. The results also show that the effect of inlet flow angle on the oscillations of mixed convection is evident, the configuration with φ = 0° is the most unstable among the four values of φ. The non-periodic oscillatory solution at Re = 2500 is studied by means of phase portraits, correlation dimension, Kolmogorov entropy and Lyapunov exponents to detect chaos. The phase portraits show the evolution of the attractor from a stable fixed point to a limited cycle to chaos, and finally, to a stable fixed point again, and the correlation dimension, Kolmogorov entropy and the largest Lyapunov numbers all show that the behavior of mixed convection in this dynamical system lies on a low-dimensional chaotic attractor according to the non-periodic oscillatory solution.     

Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

Rich and complex buoyancy-driven flow field due to natural convection will be studied numerically over a wide range of Rayleigh numbers in a cubic cavity by virtue of the simulated bifurcation diagram, limit cycle, power spectrum and phase portrait. When increasing the Rayleigh number, the predicted flow is found to evolve from the conductive state to the state with the onset of convection, which is featured with the steady and symmetric laminar solution, and then to the asymmetric state (pitchfork bifurcation), which will not be discussed in this paper. As the Rayleigh number was further increased, a limit cycle branching from the fixed point of the investigated dynamical system is observed. Supercritical Hopf bifurcation is confirmed to be the birth of the orbitally stable limit cycle that separates the vortex flow into an inner unstable region (moving away from the vortex coreline) and an outer stable region (moving towards the vortex coreline). As the Rayleigh number is increased still, the investigated buoyancy-driven flow became increasingly destabilized through quasi-periodic bifurcation and then through two predicted frequency-doubling bifurcations. Thanks to the power spectrum analysis, bifurcation scenario was confirmed to have an initially single harmonic frequency, which is featured with a driving amplitude. Then an additional ultraharmonic frequency showed its presence. Prior to chaos, in the five predicted arithmetically related frequencies there exists one frequency that is incommensurate to the other two fundamental frequencies. This computational study enlightens that the investigated nonlinear system, which involves frequency-doubling bifurcations, loses its stability to a quasi-periodic bifurcation featured with the formation of a subharmonic frequency. Subsequent to the formation of three frequency-doubling bifurcations and one quasi-periodic bifurcation, an infinite number of frequencies was observed in flow conditions with the continuously increasing Rayleigh numbers. Finally, the chaotic attractor was predicted to be evolved from the strange attractor in the corresponding phase portraits.     

Rayleigh–Benard convection in rotating fluids

Linear and weakly nonlinear properties of Rayleigh–Benard convection in rotating fluids are investigated. Linear stability analysis is studied to investigate analytically the effect of Coriolis force on gravity-driven convection for idealised stress-free boundary conditions. We have derived a nonlinear one-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation. A coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at the supercritical Hopf bifurcation are derived and discussed the stability regions of travelling and standing waves.     

Nonlinear thermohaline convection in rotating fluids

Linear and weakly nonlinear properties of thermohaline convection in rotating fluids are investigated. Linear stability analysis is studied by plotting graphs for different values of physical parameters relevant to the Earth’s outer core and oceans. We have derived a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients near the onset of stationary convection at the supercritical pitchfork bifurcation and shown the occurrence of Eckhaus and zigzag instabilities. We have studied heat transfer by using Nusselt number which is obtained from Landau–Ginzburg equation at the onset of stationary convection for the steady case. A coupled two-dimensional Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection are derived and the stability regions of travelling and standing waves discussed.     

Transition from natural to mixed convection for steam–gas flow condensing along a vertical plate

An analysis method based on two-phase boundary layer analysis has been developed to study the effects of superimposed forced convection on natural convection steam–gas flow condensing along a vertical plate. The mechanism by which superimposed forced convection enhances heat transfer is evaluated: the bulk flow blows away non-condensable gases accumulating near the interface, resulting in an elevated condensation driving force. Further, this bulk flow blowing capability may be characterized by a conventional mass transfer driving potential. Results of the new model are shown to be consistent with experimental data. Finally, a simple criterion was developed to identify transition to mixed convection from natural convection steam–gas flow.     

Loss of axisymmetry in the mixed convection, assisting flow past a heated sphere

For flows presenting linear instabilities, the laminar regime can be delimited accurately by the onset of the first bifurcation. For an unheated sphere, the primary bifurcation is preceded by a detachment of the boundary layer and a build up of a recirculation zone. In the mixed convection, the convection tends to prevent the boundary layer of the assisting flow from detaching and from a build up a recirculation zone. In this paper, the issue of the correlation between the boundary layer detachment and the loss of axisymmetry in the assisting flow is investigated with a special focus to the Prandtl number corresponding to flows in air (Pr = 0.72). For Richardson numbers up to 0.7, the detachment of the boundary layer (not necessarily a build up of the recirculation zone) is shown to be a precursor sign of a regular primary bifurcation similarly as for the wake of an unheated sphere. At this bifurcation, the flow stays steady but looses its axisymmetry. To assess the Prandtl number effects, the separation of the axisymmetric flow is investigated also in the Pr = 7 parameter plane and for Pr varying between 0.1 and 100 at fixed Reynolds and Richardson number values. The interest of the obtained results is twofold. Firstly, in the investigated parameter sub-domain, clear limits of the physical relevance of axisymmetric computations have been found. Within these limits, accurate values of the drag coefficients and overall Nusselt numbers are given. Secondly, in axisymmetric simulations, the detachment of the boundary layer may be a useful indication of the possible loss of axisymmetry.     

Linear and nonlinear stability of Soret‐Dufour Lapwood convection near double codimension‐2 points

In this study, the Dufour and Soret effects on natural double‐diffusion convection in a horizontal porous layer was studied numerically using FORTRAN 90 programming and analytically near various convection onset thresholds. The porous layer was subject to a uniform heat and mass fluxes on the horizontal walls while the vertical walls were impermeable and adiabatic. The Darcy model along with the Boussinesq approximation was assumed in the problem formulation. The governing parameters of the problem are the thermal Rayleigh number, R_T, the buoyancy ratio, N, the Lewis number, Le, the aspect ratio of the cavity, A, and the Dufour, D_u, and Soret, S_r, numbers. For a shallow enclosure, the analytical solution was derived assuming zero convection wave number, which is valid near and above criticality. The onset of subcritical, supercritical and oscillatory convection was investigated. Two linear and nonlinear codimension‐2 points were found to exist. Whether the system was subject to constant fluxes and heat and solute, regardless of the aspect ratio of the layer, the subcritical convection behavior remained the same with similarity in the thresholds expressions for subcritical bifurcation.     

Hydromagnetic mixed convection stagnation flow with suction and blowing

This paper deals with steady, two-dimensional, mixed convection flow of an electrically-conducting and heat-absorbing fluid near a stagnation point on a semi-infinite vertical permeable surface at arbitrary surface heat flux variations in the presence of a magnetic field. Similarity equations are derived and solved numerically by an implicit and accurate finite-difference method. Graphical solutions for the local skin-friction coefficient and the local Nusselt number are presented and discussed for various parametric conditions. These results are presented to illustrate the influence of the Hartmann number, wall mass transfer coefficient, heat absorption coefficient, Prandtl number and the mixed convection or buoyancy parameter.     

Stability analysis of MHD radiative mixed convective flow in vertical cylindrical annulus: Thermal nonequilibrium approach

In the present study, the influence of the induced magnetic field on the MHD mixed convective electrically conducting fluid flow inside the vertical cylindrical annulus is analyzed numerically. The heat transfer is presumed to be due to a combination of mixed convection and radiation. The stability of the flow is examined when the solid and fluid phases are not in local thermal equilibrium. The governing equations are solved numerically by both finite difference and finite element methods. To control the flow formation rate more accurately the induced magnetic field is also considered in this study. As the magnetic Prandtl number (Pm) and Hartmann number (M) get enhanced, the velocity and induced magnetic fields get retarded in the annulus due to the presence of drag-like force, namely, the Lorentz force. When there is an increase in the mixed convection parameter the induced magnetic field gets enhanced. An increase in radiation parameter tends to decline the fluid temperature and reverse the behavior of the solid temperature. Increment in Pm decreases the wall shear stress near the conducting cylinder. Increasing values of porous, magnetic, and radiation parameters lead to an unstable system with smaller heat transfer coefficient values but the system gets stabilized for larger values of heat transfer coefficient. The results could be used as first-hand information for comprehending and developing the thermal flow phenomenon in porous media. The obtained numerical results are in good accordance with the existing results. Using an artificial neural network, heat transfer characteristics are analyzed through mean square error and regression analysis.     

Investigation of torsional instability, bifurcation, and chaos of agenerator set

In this paper, the nonlinear phenomenon known as Hopf bifurcation, chaos and asynchronous operation of a simple power system are explored. Firstly, taking into account the nonlinearity of the generator shaft and the interaction of mechanics and electrics in the generator sets, the authors obtain a transient model by combining Park equations and mechanics equations. Then the Hopf bifurcation, period-doubling bifurcation and chaos caused by too large a line resistance are investigated with nonlinear mode and Floquet theory. The bifurcation figure of the system is also given. Further study shows that the chaos attractor breaks up into asynchronous operation when the resistance becomes larger. This way of loss-of-stability is different from that caused by loss-of-excitation     

FREE CONVECTION IN OPEN-TOP ENCLOSURES FILLED WITH A POROUS MEDIUM HEATED FROM BELOW

Free convection is studied for porous medium-filled enclosures that are open for fluid flow at the top. For such setups a mixed boundary condition for the transport variable at the top is examined, which is different from the classical approach (Lapwood problem) in systems where flow is governed by Darcy's law. While the latter led to open and to closed paths within each convection cell, the mixed boundary condition induces open convection cells only. By numerical means, the onset of convection, total heat and mass transfer, and the transition from the first to the second mode are examined. At 16.5, the critical Rayleigh number for the onset of convection in the system with mixed boundary condition is much lower than the classical value.     

Heat transfer of couple stress fluid from a vertical funnel: Impedance boundary conditions

The present level of literature on the subject matter indicates that nothing is known on the heat transfer across the couple stress rheological fluid flowing over a vertical avenue with Robin (mixed) wall conditions. The obtained conservation equations of the model are solved through DTM (differential transform method) and RPM (regular perturbation method). The nondimensional parameters obtained are a couple stress parameter, Brinkman number, mixed convection parameter, and Biot number. The computations reveal that flow acceleration and thermal enhancement is induced with increasing mixed convection parameter and Brinkman number for symmetric and asymmetric conditions. Increasing couple stress parameters dwindle the velocity and temperature for symmetric and asymmetric cases. The large values of the mixed convection parameter and Brinkman number increase the Nusselt values at the left wall and reduces at the right wall. The mass flow rate is augmented with the mixed convection parameter and Brinkman number but it is reduced with the couple stress parameter. The DTM, RKSM, and RPM solutions are in good agreement.     

Effects of Heating Location and Size on Natural Convection in Partially Heated Open-Ended Enclosure by Using Lattice Boltzmann Method

Natural convection heat transfer in an square enclosure, consisting of a partially heated west wall with east end open to ambient, is investigated numerically by using an in-house computational fluid dynamics solver based on thermal lattice Boltzmann method. In particular, the influences of Rayleigh number (10³–10⁶), heating location (bottom, middle, and top) on west wall, and dimensionless heating length (0.25–0.75) on momentum and heat transfer characteristics of air are presented and discussed. The streamline patterns show bifurcation at the lowest Rayleigh number for bottom and middle heating, whereas at the highest Rayleigh number, all heating positions yield bifurcation and elongation of flow patterns with a secondary vortex near the lower side of open end. The middle and bottom heating locations show a linear increase in Nusselt number with heater size, whereas inverse dependence is seen for top heating. The maximum heat transfer is observed in the case of middle heating. As expected, average Nusselt number increased with increasing Rayleigh number. Finally, the functional dependence of the average Nusselt number on flow governing parameters (Rayleigh number and heating length) for different heating locations is presented as a simple predictive empirical relationship.     

Numerical analysis of transient mixed convection flow in storage tank: influence of fluid properties and aspect ratios on stratification

Sensible heat storage in fluids generates thermal stratification. In order to improve thermodynamic system efficiency, stratification should be promoted much more. To this scope, this article presents a numerical study of transient mixed convection. The study investigates the use of different fluids as a heat storage medium in cylindrical cavities with different aspect ratios. The effect of the fluids is made through the variation of physical properties represented through the Prandtl number. The system consists of a cavity with fluid injection at the top and fluid discharge at the bottom. Transient, two-dimensional, mixed convection flows in a thermal storage tank have been studied using finite volume method. The governing equations are the conservation equations for laminar natural convection flow based on the Boussinesq approximation. Forced convection flow is superimposed through the use of appropriate boundary conditions (inflow and outflow conditions). The study considers three representative fluids i.e. Torada oil, ethylene glycol and water. It considers also cavities with aspect ratios varying from 3 to 1/3. Flow analysis is made through typical transient temperature distributions for the three fluids and for different configurations. The performances of thermal energy storage using these fluids are analyzed through the transient thermal storage efficiency.     

From natural to mixed convection in horizontal and differentially heated annular ducts: Linear stability analysis

Linear stability analysis of a fully developed mixed convection flow of air in an annular horizontal duct is numerically investigated for the radius ratio R = 1.2, a Péclet and a Rayleigh number less than 200 and 6000, respectively. An iterative method is developed to enable the convergence of the dimensionless parameters to their marginal values at the transition. New mixed convection flows are highlighted that are highly correlated with those obtained in natural convection problems under the assumption of two dimensionality. The synthesis of our results on the transitions permits us to build the map of stability for the steady and established mixed convection flows and clearly shows the occurrence of multiplicity of solutions for some couples of Rayleigh and Péclet numbers.     

Mixed convection flow over a stretching sheet of variable thickness: Analytical and numerical solutions of self‐similar equations

In this study, a mixed convection flow over a nonlinearly stretching sheet of variable thickness is examined. Governing equations are modeled and transformed into dimensionless forms by utilizing dimensionless variables. For further investigation, dimensionless, coupled nonlinear differential equations with suitable boundary conditions are numerically solved using the Matlab built‐in function bvp5c tool, and analytical solutions are also computed using the homotopy analysis method. A comparative study is carried out to check the efficiency and accurateness of the proposed solution methodologies. Convergence of the derived series solutions is carefully checked. The impact of wall thickness parameter, velocity index parameter, Prandtl number, and mixed convection parameter on nondimensional velocity, temperature, skin friction coefficient, and local Nusselt number is examined. The novelty of this examination is that the dimensionless equations are self‐similar in the presence of mixed convection. These self‐similar equations are acquired by establishing a relationship between velocity and temperature power index parameters, and similarity solutions exist only for a particular form of variable surface temperature.     

Magnetohydrodynamic mixed convection in a non-Newtonian third-grade fluid flowing through vertical parallel plates: A semianalytical study of flow and heat transfer

Buoyancy assisted and buoyancy opposed mixed convection of a third-grade fluid, which flows through vertically oriented parallel plates, subjected to uniform and constant wall heat fluxes, under the effect of an externally applied magnetic field, are investigated. The coupled, nonlinear conservation equations of momentum and energy are solved employing the collocation method (CM) and velocity and temperature distributions are solved semianalytically. The results produced by the CM and the results of exact solution are compared for the buoyancy assisted and buoyancy opposed flow of a Newtonian fluid through the vertically oriented parallel plates arrangement without the effect of the externally applied magnetic field. An excellent agreement is exhibited by demonstrating the efficacy of the CM. The effects of the third-grade fluid parameter, Hartmann number, and mixed convection parameter on the dimensionless velocity, temperature, and Nusselt number are studied. The results imply that in the case of buoyancy assisted flow, an increment in the non-Newtonian third-grade fluid parameter causes a decrease in the fluid velocity near the plate walls, which finally causes an increase in the velocity in the central core of the plates. In buoyancy opposed flow, the effect of the same parameter is to oppose the flow reversal near the walls and with higher values of this parameter, it can totally prevent the flow reversal near the walls. The results of the present study can be useful in the fields of flow and heat transfer of various grades of polymers, paints, and food processing.     

The second largest Lyapunov exponent and transition to chaos of natural convection in a rectangular cavity

In this study we have proposed an accurate and simple method to evaluate the Lyapunov spectrum. The method is suitable for any discretization method that finally expresses a governing equation system in the form of an ordinary differential equation system. The method was applied to evaluate up to the second largest Lyapunov exponents for natural convection in a rectangular cavity with heated and cooled side walls. The main results are as follows: (1) the largest and second largest Lyapunov exponents can be evaluated without any parameters that affects the exponents. (2) The second largest Lyapunov exponent makes it possible to classify quantitatively thermal convection fields into five regimes against the Rayleigh number and to clarify the transition route from steady state to chaos by identifying the first and second Hopf bifurcations. (3) The fluctuation in thermal convection fields just over the critical Rayleigh number at which Hopf bifurcation occurs can be quantitatively explained by using normalized Lyapunov vectors, associated with the computation of the Lyapunov exponents, just under the critical point.     

MHD mixed convection flow adjacent to a vertical plate with prescribed surface temperature

The steady MHD mixed convection flow adjacent to a bounding surface immersed in an incompressible viscous fluid is considered. The governing system of partial differential equations is first transformed into a system of ordinary differential equations, before being solved numerically by a finite-difference scheme. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity and temperature profiles. It is found that dual solutions exist for both assisting and opposing flows. The range of the mixed convection parameter for which the solution exists increases in the presence of a magnetic field.     

MAGNETOHYDRODYNAMIC MIXED CONVECTION FROM A VERTICAL PLATE EMBEDDED IN A POROUS MEDIUM

Magnetohydrodynamic mixed convection flaw about a vertical flat plate embedded in a porous medium is considered. The effect of the magnetic field strength on the local Nusselt number and local wait shear stress is presented. The non-Darcian model including both the inertia and boundary effects is used. A particular transformation for the governing equations is adopted to cover the whole mixed convection regime within two finite limits. Appreciable effects of the magnetic field strength on the local Nusselt number as well as on the local wall shear stress in the mixed convection regime are found.