R.A.W.M. Henkes

Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models

The laminar and turbulent natural-convection flow in a two-dimensional square cavity heated from the vertical side is numerically calculated up to a Rayleigh number of 1014 for air and up to 1015 for water. Three different turbulence models are compared: the standard k-e model with logarithmic wall functions and the low-Reynolds-number models of Chien, and Jones and Launder. The position of the laminar-turbulent transition in the vertical boundary layer strongly depends on the turbulence model used. Moreover, multiple solutions for the transition position can occur for a fixed Rayleigh number at the same numerical grid. The thermal stratification in the core of the cavity breaks up when the flow becomes turbulent. Comparison of the averaged wall-heat transfer with experiments for the hot vertical plate and for tall vertical cavities shows that the standard k-e model gives a too high prediction, whereas the low-Reynolds-number models are reasonably close to the experiment.

Numerical study of laminar and turbulent natural convection in an inclined square cavity

Two-dimensional numerical simulations of the natural convection flow of air in a differentially heated, inclined square cavity were performed for both laminar and turbulent flows. The angle of inclination of the cavity was varied from 0° (heated from below) to 180° (heated from above). For Rayleigh numbers between 104 and 1011 the natural convection flow has been calculated. A detailed analysis was made for Rayleigh numbers of 106 and 1010. The standard k−e model for turbulence was used in the prediction of turbulent flows. Numerical predictions of the heat flux at the hot wall and the influence of the angle of inclination on the Nusselt number are presented. The Nusselt number shows strong dependence on the orientation of the cavity and the power law dependence on the Rayleigh number of the flow. Flow patterns and isotherms are shown to give greater understanding of the local heat transfer. For the high Rayleigh number calculations hysteresis of the solution was found at a transition of flow patterns.

Transition to time-periodicity of a natural-convection flow in a 3D differentially heated cavity

Abstract The steady and time-periodic flow of air in a differentially heated cubical cavity has been studied numerically, using the finite-volume method. In the steady flow regime, the scaling in the boundary layer along the wall has been investigated. In the periodic flow regime, the calculated frequency was almost the same as for the two-dimensional square cavity, suggesting that the same instability mechanism is in both cases responsible for the bifurcation. There was, however, a strong three-dimensionality in the distribution of the amplitude of the oscillations.

Comparison of turbulence models for the natural convection boundary layer along a heated vertical plate

Abstract With a numerical code for solving the boundary-layer equations, the performance of different turbulence models for the natural convection boundary layer for air along a heated vertical plate is tested. The algebraic Cebeci-Smith model, the standard k-e model with wall functions for k and e and different low-Reynolds number k-e models are tested. The Cebeci-Smith model calculates a too low wall-heat transfer and turbulent viscosity. The standard k-e model with wall functions gives a too high wall-heat transfer, but the velocity and temperature profiles agree reasonably with experiments. Accurate wall-heat transfer results require the use of low-Reynolds number k-e models; the models of Lam and Bremhorst, Chien, and Jones and Launder perform best up to a Grashof number of 1011. For larger Grashof numbers the Jones and Launder model is best. A sensitivity study shows that the wall-heat transfer with the standard k-e model largely depends on the choice of the wall functions for k and e. Replacing these wall functions by zero wall conditions for k and e and adding the functions D and ƒ μ of the Chien model to the standard k-e model gives a simple, but accurate low-Reynolds number k-e model for the natural convection boundary layer.

The Reynolds-stress model of turbulence applied to the natural-convection boundary layer along a heated vertical plate

Abstract The turbulent natural-convection boundary layer for air along a heated vertical plate is investigated numerically with an algebraic (ASM) and fully differential Reynolds-stress model (RSM). From the literature a set of model constants is selected, in such a way that the wall-heat transfer and mean-flow structure are predicted in close agreement with the experimental data. Sensitivity tests on RSM constants show which constants dominate the mean-flow prediction, and which constants only affect turbulence quantities. Wall modifications are employed to improve predictions of the near-wall turbulence. RSM calculations of the turbulence quantities agree well with available experimental data. ASM results are poorer, but still in qualitative agreement with experiments. Hence, in natural-convection boundary layers, the local-equilibrium assumption has only limited applicability. Furthermore, the eddy-viscosity concept used in the k−e model (KEM) is tested. The KEM gives good mean-flow results, but for a good prediction of the detailed turbulence structure the RSM is needed.

Scaling of the laminar natural-convection flow in a heated square cavity

Abstract The steady laminar natural-convection flow of air and water in a square heated cavity is calculated for increasingly large Rayleigh number. The flow is calculated by solving both the Navier-Stokes equations and the boundary-layer equations. The results are used to determine the proper scalings of the flow in the different asymptotic flow regions: vertical boundary layers, core region, corner region and horizontal boundary layers. In particular the scalings according to the Navier-Stokes equations agree with the asymptotic model for the core and vertical boundary layers as proposed by Gill [ J. Fluid Mech. 26 , 515–536 (1966)].

Numerical determination of wall functions for the turbulent natural convection boundary layer

In an analytical way George and Capp (Int. J. Heat Mass Transfer22 813–826 (1979)) and Cheesewright (“The scaling of turbulent natural convection boundary layers in the asymptotic limit of infinite Grashof number”, paper presented at Euromech Colloquium 207 (1986)) have derived wall functions for the natural convection boundary layer along a heated vertical plate. These wall functions are compared here with numerical calculations for air, using a k-e turbulence model with low-Reynolds number modifications. George and Capps wall function for the temperature in the inner layer agrees with the calculations, but their wall function for the velocity does not. George and Capps defect laws for the velocity and the temperature in the outer layer are also numerically found, but Cheesewrights wall function for the velocity in the (lower part of the) outer layer does not agree.

Laminar natural convection boundary-layer flow along a heated vertical plate in a stratified environment

Abstract All similarity solutions of the laminar natural convection boundary-layer equations for air are numerically determined for a fixed wall and variable environment temperature. It is found that the positive M class does not have the singularity found by Merkin for a variable wall and fixed environment temperature. Solutions of the negative M class for an unstable stratification depend on the position of the outer edge and are unusable. The similarity solutions for a stable stratification show regions of backflow. Therefore, the calculation of non-similar solutions of the boundary-layer equations along a heated vertical plate with a sharp leading edge requires that the solution is known at the end of the plate. The positive M class provides such a solution for a semi-infinite plate. If the environment temperature becomes equal to the wall temperature at a finite distance x 0 , the non-similar solution does not smoothly approach the negative M class similarity solution close to x 0 .

Quantitative infrared-thermography for wall-shear stress measurement in laminar flow

Abstract The hot film is a common technique to measure wall-shear stress in boundary-layer flows. A new technique to measure the wall-shear stress, referred to as quantitative infrared-thermography, has been developed. It replaces the internal heating and the temperature detection using the hot film with the external heating using a laser and the external temperature measurement using an infrared camera, respectively. First the laser creates a hot spot on a substrate of polycarbonate covered with a plastic foil; the conductivity of the chosen substrate is small and the emissivity is large, giving the required small, but clearly detectable spot. Then the laser is switched off and the temperature decay is monitored. The measured unsteady wall temperature is used as a boundary condition to numerically solve the heat transfer in the solid, which gives, through the total heat balance, the heat transfer from the solid to the fluid. A local similarity relation, which applies for small spots, is used to relate the wall-heat transfer to the wall-shear stress. The technique is demonstrated for the Blasius boundary layer in a wind-tunnel experiment, where an accuracy of about 10% has been achieved.