R. Kahawita

NUMERICAL COMPUTATION OF THE NATURAL CONVECTION FLOW ABOUT A HORIZONTAL CYLINDER USING SPLINES

The present work is devoted to the numerical study of laminar natural convection flow from a heated horizontal cylinder under diverse surface boundary conditions using the spline fractional step method. A general formulation to treat mixed boundary conditions using the spline approximation has been presented. Numerical solutions have been obtained by solving the Navier-Stokes and energy equations. The results for the isothermal boundary condition as well as for the uniform heat flux are in good agreement with published experimental data and with other solutions presently available in the literature. Some new computations at very high Rayleigh numbers indicate the existence of attached separation vortices in the downstream plume region, the appearance of these vortices being dependent on the values of the Biot number. All results were computed on a personal computer using unequally spaced grids that provided good results with a minimum number of computational points. The numerical scheme presented here app...

Calibrating DRASTIC using field measurements, sensitivity analysis and statistical methods to assess groundwater vulnerability

The objective of this research is to indicate how the vulnerability potential to pollution may be assessed more accurately by correcting both the ratings and weights of the parameters used in the popular EPA DRASTIC method. In order to demonstrate the applicability of the proposed method, the Astaneh Aquifer located in northern Iran was selected. Nitrate concentrations were used to correlate the pollution potential in the aquifer to the DRASTIC index. The results indicated that the modified DRASTIC was significantly superior to the original method.

Modelling wetting and drying effects over complex topography

The numerical simulation of free surface flows that alternately flood and dry out over complex topography is a formidable task. The model equation set generally used for this purpose is the two-dimensional (2D) shallow water wave model (SWWM). Simplified forms of this system such as the zero inertia model (ZIM) can accommodate specific situations like slowly evolving floods over gentle slopes. Classical numerical techniques, such as finite differences (FD) and finite elements (FE), have been used for their integration over the last 20–30 years. Most of these schemes experience some kind of instability and usually fail when some particular domain under specific flow conditions is treated. The numerical instability generally manifests itself in the form of an unphysical negative depth that subsequently causes a run-time error at the computation of the celerity and/or the friction slope. The origins of this behaviour are diverse and may be generally attributed to: 1. The use of a scheme that is inappropriate for such complex flow conditions (mixed regimes). 2. Improper treatment of a friction source term or a large local curvature in topography. 3. Mishandling of a cell that is partially wet/dry. In this paper, a tentative attempt has been made to gain a better understanding of the genesis of the instabilities, their implications and the limits to the proposed solutions. Frequently, the enforcement of robustness is made at the expense of accuracy. The need for a positive scheme, that is, a scheme that always predicts positive depths when run within the constraints of some practical stability limits, is fundamental. It is shown here how a carefully chosen scheme (in this case, an adaptation of the solver to the SWWM) can preserve positive values of water depth under both explicit and implicit time integration, high velocities and complex topography that may include dry areas. However, the treatment of the source terms: friction, Coriolis and particularly the bathymetry, are also of prime importance and must not be overlooked. Linearization with a combination of switching between explicit–implicit integration can overcome the ‘stiffness’ of the friction and Coriolis terms and provide stable numerical integration. The treatment of the bathymetry source term is much more delicate. For cells undergoing a transient wet–dry process, the imposition of zero velocity stabilizes most of the approximations. However, this artificial zero velocity condition can be the cause of considerable error, especially when fast moving fronts are involved. Besides these difficulties linked with the internal position of the front within a cell versus the limited resolution of a numerical grid, it appears that the second derivative that defines whether the bed is locally convex or concave is a key indicator for stability. A convex bottom may lead to unbounded solutions. It appears that this behaviour is not linked to the numerics (numerical scheme) but rather to the mathematical theory of the SWWM. These concerns about stability have taken precedence, until now, over the crucial and related question of accuracy, especially near a moving front, and how these possible inaccuracies at the leading edge may affect the solution at interior points within the domain. This paper presents an in depth, fully two-dimensional space analysis of the aforementioned problem that has not been addressed before. The purpose of the present communication is not to propose what could be viewed as a ‘final solution’, but rather to provide some key considerations that may reveal the ingredients and insight necessary for the development of accurate and robust solutions in the future.

TRANSIENT LAMINAR NATURAL CONVECTION FROM HORIZONTAL CYLINDERS

Abstract The unsteady laminar natural convection flow from a heated horizontal cylinder under diverse surface boundary conditions is investigated numerically using the spline fractional step method. Some characteristics of the boundary layer obtained with a scale analysis are compared with the numerical results. The development of the plume region as well as the surface heat transfer and local flow field are evaluated. At small times, the present numerical solutions approach the boundary layer results and are in good agreement with the results from the scale analysis. A more detailed study of the development of the plume region, using computed particle trajectories is reported. All results are obtained using a personal computer. Qualitative comparisons between the present results and flow visualization experiments partially verify the numerical results.

Melting of ice in a porous medium heated from below

Abstract A numerical study is made of the melting of ice heated from below in a cavity filled with a porous medium, using the time-dependent form of the governing equations. The natural convection in the liquid phase, the conduction in the subcooled ice region, and the effect of density inversion of water are incorporated into the numerical simulation. The primary characteristics of the melting process, i.e. the onset of convection, the flow pattern in the melt, the heat transfer rate, and the interface position, are studied in terms of the Rayleigh number, the Stefan number, the aspect ratio of the cavity, and the density inversion of water. Principal findings indicate that the initial cellular pattern formed just after the onset of convection passes through several intermediate forms in its transition to a final steady state. Each change in the convection pattern is accompanied by a sudden increase in the heat transfer rate and in the displacement velocity of the solid-liquid interface. A local maximum in the heat transfer rate is exhibited shortly after the establishment of the new convection pattern.

Numerical simulation of the wake flow behind trapezoidal bluff bodies

Abstract Two-dimensional numerical simulations of the Benard-von Karman hydrodynamic instability behind trapezoidal bluff bodies has been studied using the spline method of fractional steps. For lower Reynolds numbers, about Re / Re c Re c is critical Reynolds number), numerical results confirm the experimentally observed behavior reported by Goujon-Durand et al. (Phys. Rev. E 1994; 50:308), i.e. the maximum amplitude of the velocity component oscillating with the fundamental frequency follows fairly well the scaling law A max ∼( Re − Re c ) 0.5 and the position X max ∼( Re − Re c ) −0.5 . The influence of the trapezoidal shape on the value of the critical Reynolds number and on the vortex shedding is briefly discussed. It appears that the influence of the trapezoidal height H is the dominant influence on the value of Strouhal number when compared with the effect of the smaller trapezoidal base width B .

Modelling the fate of pollutant in overland flow

A physically based numerical model is developed to simulate pollutant dissolution and transport in overland flow with infiltration effects. The numerical simulation of the complete process requires the solution of the unsteady, two-dimensional St. Venant equation, an equation for the infiltration process and the pollutant transport equation. The solubility rate equation proposed in this study is used to model the dissolution of a specified spatial distribution of solid pollutant into the liquid phase. The set of equations governing the complete system are integrated numerically using various finite difference algorithms. The model may be used to predict the ultimate fate of a surface applied solid pollutant by evaluating the proportions that are dissolved, infiltrate into the subsurface or are washed-off. The hydraulic portion of the model has been validated against published field data with satisfactory agreement. Verification of the complete model, however, is based on a comparison with limited laboratory test data. Proper validation would require extensive experimentation, preferably in the field which is, up to now, unavailable.

Numerical simulation of Buoyancy-Marangoni convection in two superposed immiscible liquid layers with a free surface

Abstract Buoyancy-Marangoni convection in a cavity with side heating has been studied analytically and numerically in superposed immiscible liquid layers with a free surface. The analytical results (based on an assumption of infinite aspect ratio) indicate that four different flow patterns are possible and that these results may be anticipated on the basis of the introduction of a new parameter which represents the combined effects of Marangoni forces acting at the interface between the two liquids and at the free surface. It is shown further that the new parameter is a unique thermocapillary quantity which influences the convection in the lower layer. For finite cavities, some numerical results on the mutual influence of the two layers have been presented. The numerical results obtained near the centre of the cavity are in good agreement with the results from the analytical model for sufficiently large aspect ratios.

Oscillatory behaviour in buoyant thermocapillary convection of fluid layers with a free surface

Abstract Oscillatory behaviour in thermocapillary convection with buoyancy forces has been studied numerically for superposed immiscible liquid layers with a free surface, in which the lower layer consists of low-Prandtl-number fluid. Numerical solutions to the complete two dimensional Navier-Stokes and energy equations have been obtained using the spline integration method. Attention has been focused on flow instabilities of an oscillatory nature which appear to be induced by the buoyancy forces. An attempt to understand the origin of these instabilities and indications on how to reduce or even avoid them is made. The numerical results demonstrate that oscillatory flow in a single layer of low-Prandtl number fluid may transform to a steady state after encapsulation with a fluid of higher Prandtl number, even in the absence of Marangoni forces, except when the buoyancy and the viscous forces in the upper layer are very small when compared with the lower one. The numerical experiments also demonstrate that the addition of the combined Marangoni forces to the gravitational convection plays an important role in suppressing oscillatory flow.

A numerical study of variable density flow and mixing in porous media

A numerical study of a negatively buoyant plume intruding into a neutrally stratified porous medium has been undertaken using finite different methods. Of particular interest has been to ascertain whether the experimentally observed gravitational instabilities that form along the lower edge of the plume are reproduced in the numerical model. The model has been found to faithfully reproduce the mean flow as well as the gravitational instabilities in the intruding plume. A linear stability analysis has confirmed the fact that the negatively buoyant plume is in fact gravitationally unstable and that the stability depends on two parameters: a concentration Rayleigh number and a characteristic length scale which is dependent on the transverse dispersivity.