P. Le Quéré

Computation of natural convection in two-dimensional cavities with Chebyshev polynomials

Abstract A semi-implicit spectral method for the computation of buoyancy induced two-dimensional flow of a viscous incompressible fluid is presented. The incompressible Navier-Stokes equations expressed in terms of the primitive variables velocity and pressure, and an additional equation for the temperature, are considered. All the variables are expanded in double truncated series of Chebyshev polynomials. The pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field. The algorithm is applied to the classical problem of natural convection of a Boussinesq fluid within a differentially heated vertical square cavity. Results show that, for a Prandtl number of 0.71 corresponding to air and up to Rayleigh numbers of 106, very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction. New interesting phenomena are shown for higher values of the Rayleigh number.

Development of a local subgrid diffusivity model for large-eddy simulation of buoyancy-driven flows: Application to a square differentially heated cavity

We present a local subgrid diffusivity model for the large-eddy simulation of natural-convection flows in cavities. This model, which does not make use of the Reynolds analogy with a constant subgrid Prandtl number, computes the subgrid diffusivity independently from the subgrid viscosity along the lines of the mixed scale model for eddy viscosity. First, an a-priori test is performed from a direct numerical simulation (DNS) approach in order to compare the respective effects of the subgrid viscosity model and that introduced by the QUICK scheme used to discretize the convective terms in the momentum equations. Then the subgrid diffusivity model is applied to the case of a two-dimensional square cavity filled with air for a Rayleigh number of 5 2 10 10 . Comparisons with DNS reference results demonstrate significant improvements in capturing the general pattern of the flow and particularly in predicting the transition to turbulence in the boundary layers as compared with Reynolds analogy results. The influence of subgrid diffusivity on the local heat transfer is also examined.

Budgets of turbulent stresses and fluxes in a vertical slot natural convection flow at Rayleigh Ra = 105and 5.4 105

Abstract A direct numerical simulation (DNS) of natural turbulent convection in a differentially heated infinite vertical slot has been computed with a mixed finite difference/Fourier code at Rayleigh numbers of R a = 1.0 10 5 and R a = 5.4 10 5 . A database containing up to second-order budgets has been collected, and the physics of the Reynolds stresses and turbulent heat fluxes is analyzed in light of the relevant conservation equations. Unusual features are the negative production terms and countergradient turbulent transport. Near-wall flow characteristics are strongly influenced by transport of quantities from the core of the “Couette flow” where most of the turbulence is produced. A fairly simple second-moment closure, relying on elliptic relaxation instead of damping functions for near wall effects, is found to perform fairly well on a global scale. Although the modeling approach is open for improvements if a term-by-term analysis is conducted.

Transition to unsteady natural convection in a tall water‐filled cavity

The transition to unsteady natural convection in a water‐filled differentially heated cavity of vertical aspect ratio 10 is studied numerically by integrating two‐dimensional Navier–Stokes equations in the Boussinesq approximation. The numerical algorithm combines a pseudospectral Chebyshev space discretization with a third‐order time‐stepping scheme. The top and bottom horizontal walls are considered to be either insulated or perfectly conducting. The critical Rayleigh number for adiabatic walls is found to be more than 30 times larger than that corresponding to perfectly conducting walls. In both cases, however, examination of the space‐time structure of the temperature and velocity fluctuations shows that the onset of unsteadiness corresponds to boundary layer instability. In the case of perfectly conducting end walls, the transition to unsteady solutions possesses the characteristic features of a supercritical Hopf bifurcation. This is used to accurately determine the critical value. The dependence of the basic oscillation period in the vicinity of the critical Rayleigh number is given. Finally, the influence of unsteadiness on the local heat transfer coefficient is shown.

A Chebyshev collocation algorithm for 2D non-Boussinesq convection

Abstract A Chebyshev collocation algorithm is developed to integrate the time-dependent Navier-Stokes equations for natural convection flow with large temperature differences. The working fluid is assumed to be a perfect gas and its thermophysical properties vary with temperature according to Sutherland laws. The governing equations do not allow for acoustic waves. The generalized Helmholtz and Uzawa operators which arise from time discretization are solved iteratively and the performances of several types of preconditioners and iterative schemes are examined. The algorithm is validated by computing almost Boussinesq flows and by comparing with previous results obtained with a finite difference algorithm. We investigate the effects of the temperature difference and of total mass contained within the cavity on the transition to unsteadiness in a cavity of aspect ratio 8. It is shown that these parameters have, indeed, a significant effect on the value of Rayleigh number at which unsteadiness is triggered. We also discuss the nature of the time-periodic solution which is obtained for Ra slightly supercritical values.

Numerical simulation of diffusion-limited PSA process models by finite difference methods

A numerical method based on finite differences is proposed for simulating pressure swing adsorption processes with intraparticle diffusion controlling. The problem has two important spatial coordinates: the axial position in the bed and the location within a particle. The method proposed keeps the two coordinate scales separate. Diffusion equations with variable main- and cross-term diffusivities are solved implicitly for the particles at each time step and the solution is expressed in terms of the bed-scale variables. The material balances on the bed can then be solved with the bed-scale variables as the only unknowns. The method is computationally efficient and validated by comparison with an exact solution derived under conditions of plug flow, constant total pressure, and with a binary Langmuir isotherm. As a full example, we treat the kinetic separation of air to produce nitrogen using a fixed bed of carbon molecular sieve. Characteristics of the specific implementation are incorporation of variable-step grids with identical volumes for the intraparticle diffusion equations and the use of the third-order QUICK scheme to discretize the convection terms in the material balances for the bed.

Numerical simulations of multiple flow transitions in axisymmetric annulus convection

We examine numerically the behaviour of the solutions of the axisymmetric Boussinesq equations in a tall, differentially heated, air-filled annulus. The numerical algorithm integrates the time-dependent equations in primitive variables and combines a pseudospectral Chebyshev spatial expansion with a second-order time-stepping scheme. The instability of the conduction regime is found to be unsteady cross-rolls. By assuming Hopf bifurcation, we can accurately determine the critical Rayleigh number. As the Rayleigh number increases, the solution is monoperiodic at first. Then it undergoes a period-doubling bifurcation. When the Rayleigh number is further increased, the solution reverts to a monocellular steady state through suberitical bifurcations with hysteresis. At even higher Rayleigh number, boundary-layer instability sets in, in the form of travelling waves. This instability has the characteristics of a supercritical Hopf bifurcation. We examine the space-time structure of the two types of unsteady solutions. We have presented the basic periods of the steady oscillations as functions of the Rayleigh number in the vicinity of the Hopf bifurcation points, and have also computed the Nusselt numbers for the various flow regimes.

Effect of Tollmien Schlichting wave on convective heat transfer in a wavy channel. Part I: Linear analysis

Abstract Hydrodynamic instabilities in wavy channels and their effect on the convective heat transfer are investigated using both linear stability analysis and integration of the time dependent Navier–Stokes and energy equations. The linear stability of the fully developed flow is determined from a generalized eigenvalue problem resulting from a Galerkin approach using divergence free Chebychev basis functions and trigonometric polynomials. Several axial periodicity lengths to geometry length ratios have been considered. For our geometry, the instability is found to set in as a Tollmien Schlichting wave, at a Reynolds number approximately equal to 90. The dynamics of the detachment and reattachment points and of temperature field for constant wall temperature are examined under the assumption of small amplitude fluctuations. It is shown that, although the average heat transfer remains almost constant, large amplitude variations of the local heat transfer coefficient can be observed, this effect is increasing with increasing Prandtl number.

Axisymmetric numerical simulations of turbulent flow in rotor stator enclosures

We present axisymmetric numerical simulations of transitional and chaotic flow regimes in rotor–stator cavities of radial aspect ratio approximately 8. These simulations are carried out using a second order time and space accurate algorithm which integrates the axisymmetric unsteady Navier–Stokes equations in stream function-azimuthal vorticity-azimuthal velocity form. Detailed flow analysis has been carried out for selected values of the rotational Reynolds number (Reθ) up to 106. At the largest value considered, computations have been performed using 4096 mesh points in the radial direction, which has required using a multi-domain decomposition algorithm implemented on a parallel machine. The limitations and consequences of the axisymmetry assumption are first discussed and checked against available experimental results. The evolutions of the instantaneous flow structure and of its first and second order statistic moments as the Reynolds number increases are discussed. It is shown that the dynamics of the flow mainly consists of travelling waves propagating in the stator and rotor boundary layers and of inertial waves in the core region, and that for moderate Reynolds numbers (Reθ≃3×105), the rotor boundary layer is almost completely steady while large amplitude fluctuations are found in the stator boundary layer. The evolution of second order moments confirms the fundamentally asymmetrical role of the boundary layers along the rotor and along the stator. A turbulent kinetic energy budget is shown which exhibits some specific features attributed to the rotation effects and to a lesser extent to the axisymmetry assumption.

Transitional regimes of natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation

The transition to unsteadiness and the dynamics of weakly turbulent natural convection, coupled to wall or gas radiation in a differentially heated cubical cavity with adiabatic lateral walls, are studied numerically. The working fluid is air with small contents of water vapor and carbon dioxide whose infrared spectral radiative properties are modelled by the absorption distribution function model. A pseudo spectral Chebyshev collocation method is used to solve the flow field equations and is coupled to a direct ray tracing method for radiation transport. Flow structures are identified by means of either the proper orthogonal decomposition or the dynamic mode decomposition methods. We first retrieve the classical mechanism of transition to unsteadiness without radiation, characterized by counter-rotating streamwise-oriented vortices generated at the exit of the vertical boundary layers. Wall radiation through a transparent medium leads to a homogenization of lateral wall temperatures and the resulting transition mechanism is similar to that obtained with perfectly conducting lateral walls. The transition is due to an unstable stratification upstream the vertical boundary layers and is characterized by periodically oscillating transverse rolls of axis perpendicular to the main flow. When molecular gas radiation is accounted for, no periodic solution is found and the transition to unsteadiness displays complex structures with chimneys-like rolls whose axes are again parallel to the main flow. The origin of this instability is probably due to centrifugal forces, as suggested previously for the case without radiation. Above the transition to unsteadiness, at Ra = 3 × 108, it is shown that both wall and gas radiation significantly intensify turbulent fluctuations, decrease the thermal stratification in the core of the cavity, and increase the global circulation.