V. Dakshina Murty

A numerical investigation of Benard convection using finite elements

Abstract The results of numerical calculations based on the finite element for the two dimensional Benard convection are presented. The fluid is confined between two horizonal solid walls and is subjected to a vertical temperature gradient. Aspect ratios of 10.0 and 10.5 are used. The effect of the hydrodynamic boundary conditions on the vertical walls on the number and the size of cells is demonstrated. Values of average Nusselt number are presented for various Rayleigh numbers ranging from 1710 to 10,000. The conduction mode of heat transfer is replaced by the convection mode at a Rayleigh number of 1710, with the latter mode becoming more prominent at higher Rayleigh numbers. The calculation of the length of each cell and the heat flux compare very well with the values reported in the literature.

A finite element solution of double diffusive convection

Abstract The finite element method is used to analyze double diffusive convection in a rectangular box. The dimensions of the domain are chosen such that it matches exactly with a unit cell. Three cases are studied which correspond to three sets of boundary conditions on the walls. Numerical results are presented for Prandtl numbers from 0.1 to 10.0 and thermal Rayleigh number ranging from 2,000 to 10,000. Solute Rayleigh number is held constant at 1,000 throughout the study. The type of element used is the eight noded element with all dependent variables except the pressure interpolated quadratically and the pressure interpolated linearly. Numerical results are presented in terms of isotherms, streamlines, and isohals. Oscillatory solutions are obtained for a certain range of thermal Rayleigh numbers from the steady state algorithm. The oscillations can be seen in terms of the numerical solution oscillating from iteration to iteration about a mean solution.

On nonisothermal flows of Bingham plastics

A numerical scheme based on the Galerkins finite element method is used to analyze nonisothermal flows of Bingham plastics. These fluids exhibit yield stresses, below which the material does not flow. This condition is enforced by introducing the bi-viscosity model in which the plug behaviour is approximated by a highly viscous fluid. The bi-viscosity model approaches the ideal Bingham plastic model if the pre-yield viscosity becomes large. This numerical method is applied to solve three problems namely the Graetz-Nusselt, recirculating flow, and sinusoidal channel flow problems for various values of the yield stress. The size of the plug as well as the velocity and temperature fields correspond closely with values available in literature.

A study of the effect of aspect ratio on Benard convection

Abstract The effect of aspect ratio on the stability of two-dimensional Benard convection is studied numerically. Five values of aspect ratio are considered; they are 2, 3, 4, 5, and 10. For a fixed Prandtl number of 1, it is found that the onset of Benard convection occurs around 1800, with the cell motion being more pronounced at this Rayleigh number for larger aspect ratios. An even number of cells appears for all the aspect ratios. The length of each cell and average Nusselt number agree well with values reported in literature.

A numerical study of the stability of thermohaline convection in a rectangular box containing a porous medium

Abstract The finite element method is used to study double diffusive convection in a rectangular box containing a porous medium. The porous medium is described by means of the Darcy-Brinkman model. The problem solved is the Benard problem in the box. It is found that the stability of the flow is dependent on a combination of thermal Rayleigh number, buoyancy ratio, and Lewis number. This combination for the onset of cellular motion can be written as Ra (1+ N.Le )=4π π 2 . This criterion holds for all combinations of Ra, N, and Le whether the thermal and solutal gradients are aiding or opposing each other. Numerical results are presented in the form of flow, temperature, and concentration fields and average Nusselt and Sherwood numbers.

Natural convection around a cylinder buried in a porous medium - non-Darcian effects

Abstract Natural convection heat transfer around a cylinder embedded in a porous medium is studied numerically using the penalty finite-element method. To model fluid flow inside the porous medium, the Brinkman and Brinkman-Forschheimer equations are used. Numerical results are obtained in the form of streamlines and isotherms. The Rayleigh number values range from 0.04 to 200. Calculated values of average heat transfer rates agree reasonably well with values reported in the literature.

A Numerical Simulation of Heat Pipe Application to the Cooling of Pulse Detonation Engines

A numerical method based on the finite elements is applied to the cooling of pulse detonation tube using heat pipe technology. Towards this end, the fluid flow and heat transfer in the wick are modeled as flow in a porous medium. The flow is described using the so called Darcy Brinkman model which has close resemblance to the Navier-Stokes equations. It is found that for Darcy numbers less than 0.0001 the results are indistinguishable from regular Darcy flows. The shape of the heat pipe is that of a fin with the proportion of the length of the evaporator section being varied. In this study two values of this ratio have been used, namely 1 and 0.5.Copyright

A Numerical Study of the Onset of Cellular Benard Convection in Shear Rate Dependent Non Newtonian Fluids in Porous Media: Non-Darcian Effects

A numerical method based on the finite element method is applied to the study of onset of Benard convection in porous media. The flow is described using the so-called Darcy Brinkman model, which has close resemblance to the Navier-Stokes equations. Itis found that for Darcy numbers less than 0.0001 the results are indistinguishable from regular Darcy flows. The non-Newtonain nature of the fluid is described by the so-called power law model, of which Newtonian fluid is a special case. Numerical results are presented for n varying from 0.4 to 1.5. The critical value of Rayleigh number for onset of convection for Newtonian fluids is found to be 40 which is close to the theoretical value of 4π2 ; boundary conditions on the horizontal walls have little effect in the sense that whether it is a slip or free (no shear condition) the results appear to be the same for onset of cellular motion. It is also found that the value of critical Rayleigh number increases with power law index.Copyright

The properties of the numerical solutions of isothermal q-f turbulence equations using finite elements

Abstract The properties of the numerical solution of turbulence equations are examined using the finite element method. The set of equations used is the q-f turbulence equations, instead of the commonly used k-e equations. It is found that from a purely numerical point of view, the former has several desirable features over the latter in that the coupling between the equations is quite weak, and the signs of diffusivities, sources and sinks are independent of the sign of q and f. An algorithm for solving the q-f equations is presented using Galerkins finite element method. Several numerical examples verifying the robustness and stability of the algorithm are presented.