Nonlinear dynamics between two differentially heated vertical plates in the presence of stratification

We consider the numerical simulation of the flow between infinite, differentially heated vertical plates with positive stratification. We use a two-dimensional Boussinesq approximation, with periodic boundary conditions in the vertical direction. The relative stratification parameter ${\gamma=(\frac{1}{4}Ra S)^{1/4}}$ , where Ra is the Rayleigh number and S the adimensional stratification, is kept constant and equal to 8. The Prandtl number is 0.71. We derive a complex Ginzburg-Landau equation from the equations of motion. Coefficients are computed analytically, but we find that the domain of validity of these coefficients is small and rely on the numerical simulation to adjust the coefficients over a wider range of Rayleigh numbers. We show that the Ginzburg-Landau equation is able to accurately predict the characteristics of the periodic solution at moderate Rayleigh numbers. Above the primary bifurcation at Ra = 1.63 × 10⁵, the Ginzburg-Landau model is found to be Benjamin-Feir unstable and to be characterized by modulated traveling waves and phase-defect chaos, which is supported by evidence from the DNS. As the Rayleigh number is increased beyond Ra = 2.7 × 10⁵, nonlinearities become strong and the flow is characterized by cnoidal waves.     

Computation of buoyancy-driven flow in an eccentric centrifugal annulus with a non-orthogonal collocated finite volume algorithm

A computational study is performed on two-dimensional mixed convection in an annulus between a horizontal outer cylinder and a heated, rotating, eccentric inner cylinder. The computation has been done using a non-orthogonal grid and a fully collocated finite volume procedure. Solutions are iterated to convergence through a pressure correction scheme and the convection is treated by Van Leer's MUSCL scheme. The numerical procedure adopted here can easily eliminate the ‘Numerical leakage’ phenomenon of the mixed convection problem whereby strong buoyancy and centrifugal effects are encountered in the case of a highly eccentric annulus. Numerical results have been obtained for Rayleigh number Ra ranging from 7×10³ to 10⁷, Reynolds number Re from 0 to 1200 and Prandtl number Pr from 0.01 to 7. The mixed rotation parameter σ (=Ra/PrRe²) varies from ∞ (pure natural convection) to 0.01 with various eccentricities ε. The computational results are in good agreement with previous works which show that the mixed convection heat transfer characteristics in the annulus are significantly affected by σ and ε. The results indicate that the mean Nusselt number Nu increases with increasing Ra or Pr but decreases with increasing Re. In the case of a highly eccentric annulus the conduction effect becomes predominant in the throat gap. Hence the crucial phenomenon on whereby Nu first decreases and then increases can be found with increasing eccentricity. © 1998 John Wiley & Sons, Ltd.     

Three-dimensional simulations of natural convection in a sidewall-heated cube

A high-resolution, finite difference numerical study is reported on three-dimensional natural convection of air in a differentially heated cubical enclosure over an extensive range of Rayleigh number from 10³ to 10¹⁰. The maximum number of grid points is 122 × 62 × 62. Solutions to the primitive variable formulation of the incompressible Navier-Stokes and energy equations are acquired by a control-volume-based procedure together with a higher-order upwind-differencing technique. The field characteristics at large-time limits are examined in detail by state-of-the-art numerical visualizations of the three-dimensional results. The emergence of the well-defined boundary layers and the interior core at high Rayleigh numbers is captured by elaborate numerical visualizations. Both the similarities and discrepancies between the three- and two-dimensional computations are pointed out. These emphasize the need for three-dimensional calculations to accurately determine the flow characteristics and heat transfer properties in realistic, high-Rayleigh-number situations.     

Numerical simulation of 2D lid‐driven cavity flow with CLEARER algorithm on extremely highly skewed grids at high Reynolds numbers

It is crucial to deal with the grid non‐orthogonality effectively in solving the flow in complex geometries, especially at high Reynolds numbers. In this study, the newly proposed Coupled and Linked Equations Algorithm Revised‐ER (CLEARER) algorithm is adopted to solve this problem successfully. In CLEARER algorithm the second relaxation factor is introduced in constructing the contravariant interface velocities, by setting it to a low value. CLEARER algorithm can overcome the severe grid non‐orthogonality and non‐linearity of equations effectively. After the numerical results with CLEARER are validated with the benchmark solutions, this algorithm is used to solve the lid‐driven flow in inclined cavity with inclination angles varying from 10 to 170^°, and Reynolds numbers varying from 5000 to 15 000. The streamlines and the centerline velocity distributions are provided in detail for all cases, which may offer some guidance for the study in this area. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental investigation of natural convection heat transfer in horizontal and inclined annular fluid layers

In the present study, an experimental investigation of heat transfer and fluid flow characteristics of buoyancy-driven flow in horizontal and inclined annuli bounded by concentric tubes has been carried out. The annulus inner surface is maintained at high temperature by applying heat flux to the inner tube while the annulus outer surface is maintained at low temperature by circulating cooling water at high mass flow rate around the outer tube. The experiments were carried out at a wide range of Rayleigh number (5 × 10⁴ < Ra < 5 × 10⁵) for different annulus gap widths (L/D _o = 0.23, 0.3, and 0.37) and different inclination of the annulus (α = 0°, 30° and 60°). The results showed that: (1) increasing the annulus gap width strongly increases the heat transfer rate, (2) the heat transfer rate slightly decreases with increasing the inclination of the annulus from the horizontal, and (3) increasing Ra increases the heat transfer rate for any L/D _o and at any inclination. Correlations of the heat transfer enhancement due to buoyancy driven flow in an annulus has been developed in terms of Ra, L/D _o and α. The prediction of the correlation has been compared with the present and previous data and fair agreement was found.     

Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions

A finite volume multigrid procedure for the prediction of laminar natural convection flows is presented, enabling efficient and accurate calculations on very fine grids. The method is fully conservative and uses second-order central differencing for convection and diffusion fluxes. The calculations start on a coarse (typically 10 × 10 control volumes) grid and proceed to finer grids until the desired accuracy or maximum affordable storage is reached. The computing times increase thereby linearly with the number of control volumes. Solutions are presented for the flow in a closed cavity with side walls at different temperatures and insulated top and bottom walls. Rayleigh numbers of 10⁴, 10⁵ and 10⁶ are considered. Grids as fine as 640 × 640 control volumes are used and the results are believed to be accurate to within 0–01%. Second-order monotonic convergence to grid-independent values is observed for all predicted quantities.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Transient natural convection between concentric and vertically eccentric spheres with mixed boundary conditions

Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with constant heat flux on the inner wall and a specified isothermal temperature on the outer wall. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. The alternating direction implicit method and the successive over-relaxation techniques are applied to solve the finite difference form of governing equations. A physical model is introduced which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field are obtained for a range of modified Rayleigh number (10³<Ra?<5×10⁵), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (ε=±0.625). Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the modified Rayleigh number and the eccentricity of the spherical annulus. The results of average Nusselt numbers are also compared with the results obtained for flow between two isothermal spheres.     

Validating the URANS shear stress transport γ −Reθ model for low‐Reynolds‐number external aerodynamics

A numerical investigation on low‐Reynolds‐number external aerodynamics was conducted using the transitional unsteady Reynolds‐averaged Navier–Stokes shear stress transport γ ?Re_θ model and the ANSYS‐CFX computational fluid dynamics suite. The NACA 0012 airfoil was exposed to chord‐based Reynolds numbers of 5.0 ×10⁴, 1.0 ×10⁵ and 2.5 ×10⁵ at 0°, 4°and 8°angles of attack. Time‐averaged and instantaneous flow features were extracted and compared with fully turbulent shear stress transport results, XFLR5 panel e ^N method results, and published higher order numerical and experimental studies. The current model was shown to reproduce the complex flow phenomena, including the laminar separation bubble dynamics and aerodynamic performance, with a very good degree of accuracy. The sensitivity of the model to domain size, grid resolution and quality, timestepping scheme, and free‐stream turbulence intensity was also presented. In view of the results obtained, the proposed model is deemed appropriate for modelling low‐Reynolds‐number external aerodynamics and provides a framework for future studies for the better understanding of this complex flow regime. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical computation of three‐dimensional incompressible Navier–Stokes equations in primitive variable form by DQ method

In this paper, the global method of differential quadrature (DQ) is applied to solve three‐dimensional Navier–Stokes equations in primitive variable form on a non‐staggered grid. Two numerical approaches were proposed in this work, which are based on the pressure correction process with DQ discretization. The essence in these approaches is the requirement that the continuity equation must be satisfied on the boundary. Meanwhile, suitable boundary condition for pressure correction equation was recommended. Through a test problem of three‐dimensional driven cavity flow, the performance of two approaches was comparatively studied in terms of the accuracy. The numerical results were obtained for Reynolds numbers of 100, 200, 400 and 1000. The present results were compared well with available data in the literature. In this work, the grid‐dependence study was done, and the benchmark solutions for the velocity profiles along the vertical and horizontal centrelines were given. Copyright © 2003 John Wiley & Sons, Ltd.     

A compact finite difference method on staggered grid for Navier–Stokes flows

Compact finite difference methods feature high‐order accuracy with smaller stencils and easier application of boundary conditions, and have been employed as an alternative to spectral methods in direct numerical simulation and large eddy simulation of turbulence. The underpinning idea of the method is to cancel lower‐order errors by treating spatial Taylor expansions implicitly. Recently, some attention has been paid to conservative compact finite volume methods on staggered grid, but there is a concern about the order of accuracy after replacing cell surface integrals by average values calculated at centres of cell surfaces. Here we introduce a high‐order compact finite difference method on staggered grid, without taking integration by parts. The method is implemented and assessed for an incompressible shear‐driven cavity flow at Re = 10³, a temporally periodic flow at Re = 10⁴, and a spatially periodic flow at Re = 10⁴. The results demonstrate the success of the method. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical and experimental studies of natural convective heat transfer from vertical and inclined narrow isothermal flat plates

Natural convective heat transfer from an isothermal narrow flat plate embedded in a plane adiabatic surface and inclined at moderate positive and negative angles to the vertical has been numerically and experimentally studied. The solution has the Rayleigh number, the dimensionless plate width, the angle of inclination, and the Prandtl number as parameters. Attention was restricted to a Prandtl number of 0.7. The numerical results have been obtained for Rayleigh numbers between 10³ and 10⁷ for dimensionless plate widths of between 0.3 and 1.2 and for angles of inclination between +45° and −45°. In the experimental study, results have been obtained for Rayleigh numbers between 4 × 10² and 10⁵ for dimensionless plate widths of 0.4 and 2.5 and for angles of inclination between +45° and −45° to the vertical. Empirical equations for the heat transfer rate have been derived.     

Sublimation of vertically oriented paradichlorobenzene cylinders in a natural convection environment

Paradichlorobenzene cylinders were cast, then suspended vertically and allowed to sublimate in air. Data on mass versus time were measured, and a sublimation rate was calculated. Three cylinders of different diameters were used: 1 inch (2.54?cm), 1.5 inch (3.81?cm), and 2 inch (5.08?cm). The length of all three cylinders was 10 in. (25.4?cm). Calculations indicate that the Schmidt number was constant. The Sherwood number ranged from 23 to 26, and Rayleigh numbers varied from 11?×?10³ to 88?×?10³. The objective of this study was to develop a correlation for determining the mass transfer coefficient of vertically suspended paradichlorobenzene cylinders in a natural convection environment. An equation relating Sherwood and Rayleigh numbers was derived.     

Experimental investigation of natural convection in a vertical rib-roughened channel with asymmetric heating

An experimental study in an open-ended vertical channel is carried out in order to describe the fluid dynamics and heat transfer of transient free convection inside a vertical rib-roughened channel asymmetrically heated at various uniform heat fluxes (650, 700, and 780 W/m²) corresponding to various modified Rayleigh numbers (3.65 × 10⁶, 3.93 × 10⁶ and 4.4 × 10⁶). Two ribs are symmetrically located on each wall. The investigations focused more specifically on the influence of the ribs positions inside the channel and the modified Rayleigh number (Ra^*) both in steady-state regime and during the transitional phase occurring just after the start of the heating on the flow structure and the heat transfer performance. The results showed the appearance of large-scale flow instabilities which will develop and propagate until the development of the pocket-like vortex (reversed flow). Also, the formation and breakup of recirculation eddies, vortex banishment, besides that a separation and shifting of the boundary layer from one wall to another are identified. The best position of the ribs for heat extraction depends on the magnitude of the Rayleigh number. In that case, the top position is the optimal position for the small and the moderate modified Rayleigh numbers.     

Parameter-free numerical method for modeling thermal convection in square cavities in a wide range of Rayleigh numbers

Some numerical results for the two- and three-dimensional de Vahl Davis benchmark are presented. This benchmark describes thermal convection in a square (cubic) cavity with vertical heated walls in a wide range of Rayleigh numbers (10⁴ to 10¹⁴), which covers both laminar and highly turbulent f lows. Turbulent f lows are usually described using a turbulence model with parameters that depend on the Rayleigh number and require adjustment. An alternative is Direct Numerical Simulation (DNS) methods, but they demand extremely large computational grids. Recently, there has been an increasing interest in DNS methods with an incomplete resolution, which, in some cases, are able to provide acceptable results without resolving Kolmogorov scales. On the basis of this approach, the so-called parameter-free computational techniques have been developed. These methods cover a wide range of Rayleigh numbers and allow computing various integral properties of heat transport on relatively coarse computational grids. In this paper, a new numerical method based on the CABARET scheme is proposed for solving the Navier–Stokes equations in the Boussinesq approximation. This technique does not involve a turbulence model or any tuning parameters and has a second-order approximation scheme in time and space on uniform and nonuniform grids with a minimal computational stencil. Testing the technique on the de Vahl Davis benchmark and a sequence of refined grids shows that the method yields integral heat f luxes with a high degree of accuracy for both laminar and highly turbulent f lows. For Rayleigh numbers up to 10¹⁴, a several percent accuracy is achieved on an extremely coarse grid consisting of 20 × 20 cells refined toward the boundary. No definite or comprehensive explanation of this computational phenomenon has been given. Cautious optimism is expressed regarding the perspectives of using the new method for thermal convection computations at low Prandtl numbers typical of liquid metals.     

Visualization of natural convection heat transfer on horizontal cylinders

Natural convection heat transfer phenomena on horizontal cylinders were investigated experimentally in order to explore the applicability of analogy experimental method using the copper electroplating system and to visualize the local heat transfer depending on the angular position and the diameter of the horizontal cylinder. The diameters of the cylinders are varied from 0.01 to 0.15 m, which correspond to the Rayleigh numbers of 1.73 × 10⁷–5.69 × 10¹¹. The measured mass transfer coefficients show good agreements with the existing heat transfer correlations. The patterns of copper plated on the aluminum cathodes for various Rayleigh numbers reveal and visualize the local heat transfer depending on the angular position and show good agreement with the works of Kitamura et al. The hydrogen bubbles produced at higher applied potential visualize the plumes appeared on top region of the cylinders.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.     

Time-dependent natural convection in a square cavity: Application of a new finite volume method

A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of 0-71 and Rayleigh numbers equal to 10³, 10⁴ and 10⁵. Both 11 × 11 and 21 × 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.