L. Elliott

A new model for viscous dissipation in porous media across a range of permeability values

In this paper a unified mathematical theory for the viscous dissipation term in the governing Brinkman equation is derived. This term has, unlike other models, the correct asymptotic behaviour in both the fully Darcy and Newtonian fluid flow limits.

An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput. Maths. Math. Phys. 31 (1991) 45] for obtaining approximate solutions to the ill-posed Cauchy problem for the Helmholtz equation is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularising criterion is also proposed.

Application of the boundary element method to inverse heat conduction problems

Abstract The solution of the one-dimensional, linear, inverse, unsteady heat conduction problem (IHCP) in a slab geometry is analysed. The initial temperature is known, together with a condition on an accessible part of the boundary of the body under investigation. Additional temperature measurements in time are taken with a sensor positioned at an arbitrary location within the solid material, and it is required to determine the temperature and the heat flux on the remaining part of the unspecified boundary. As the problem is improperly posed the direct method of solution cannot be used and hence the least squares, regularization and energy method have been introduced into the boundary element method (BEM) formulation. When noise is present in the measured data some of the numerical results obtained using the least squares method exhibit oscillatory behaviour, but these large oscillations are substantially reduced on the introduction of the minimal energy technique based on minimizing the kinetic energy functional subject to certain constraints. Furthermore, the numerical results obtained using this technique compare well with the results obtained using regularization procedures, showing a good stable estimation of the available test solutions. Further, the constraints, subject to which the minimization is performed, depend on a small parameter of which selection is more natural and easier to implement than the choice of the regularization parameter, which is always a difficult task when using the regularization procedures.

Combined Free and Forced Convection in Vertical Channels of Porous Media

In this paper we investigate the combined free and forced convection of a fully developed Newtonian fluid within a vertical channel composed of porous media when viscous dissipation effects are taken into consideration. The flow is analysed in the region of a first critical Rayleigh number in order to interpret the multiple-valued solutions and discuss their validity. The governing fourth-order, ordinary differential equation, which contains the Darcy and the viscous dissipation terms, is solved analytically using perturbation techniques and numerically using D02HBF NAG Library. A detailed investigation of the governing O.D.E. is performed on both clear fluid and porous medium for various values of the viscous dissipation parameter, ε, when the wall temperature decreases linearly with height, and the pressure gradient is both above and below its hydrostatic value. Although mathematically the results in all cases show that there are two solution branches, producing four possible solutions, the study of the velocity and buoyancy profiles together with the Darcy effect indicate that only one of the two solutions at any value of the Rayleigh number appears to be physically acceptable. It is shown that the effect of the Darcy number decreases as the critical Rayleigh numbers increase.

Development and testing of a numerical code for treatment of complex river channel topography in three-dimensional CFD models with structured grids

The potential for using high-resolution numerical modelling to understand flow over complex rough heterogeneous surfaces has yet to be fully realized, largely due to problems of designing numerically stable meshes for use with complex bed geometries. Representation of such geometries has tended to rely upon sub-grid scale treatments involving roughness parameterization within numerical schemes. This paper develops, verifies and validates a technique for dealing with the representation of complex bed geometries within a basic numerical scheme. The method is based upon a numerical porosity treatment, which uses a structured grid but specifies cell porosities to block out bottom topography (P = 1 for cells that are all water, P = 0 for cells that are all bed and 0 < P < 1 for partly blocked cells) with appropriate drag terms introduced into the momentum equations. This method has the distinct advantages of minimizing grid distortion and increasing the ease of numerical solution. A validation exercise is presented that is based upon two separate flume experiments using digital particle imaging velocimetry to collect velocity measurements of (i) flow around a cube; and (ii) a simulation of a simplified representation of bed roughness. These results are considered against simulations based upon previous treatments of complex topography, notably in relation to roughness parameterization.

Computational fluid dynamics and the physical modelling of an upland urban river

This paper describes the application of a commercially available, three-dimensional computational fluid dynamic (CFD) model to simulate the flow structure in an upland river that is prone to flooding. Simulations use a rectangular channel geometry, smooth sidewalls and a bed topography obtained from the field site that contains a subdued pool–riffle sequence. The CFD model uses the RNG κ–var epsilon turbulence closure scheme of Yakhot and Orszag (J. Sci. Comput. 1 (1986) 1), as implemented in FLUENT 4.4.4, with a free surface. Results are shown for numerical runs simulating a 1:100 year return interval flood. Output from the numerical model is compared to a physical model experiment that uses a 1:35 scale fibreglass mould of the field study reach and measures velocity using ultrasonic Doppler velocity profiling (UDVP). Results are presented from the numerical and flume models for the water surface and streamwise velocity pattern and for the secondary flows simulated in the numerical model. A good agreement is achieved between the CFD model output and the physical model results for the downstream velocities. Results suggest that the streamwise velocity is the main influence on the flow structure at the discharge and channel configuration studied. Secondary flows are, in general, very weak being below the resolution of measurement in the physical model and less than 10% of the streamwise velocity in the numerical model. Consequently, there is no evidence for a ‘velocity dip’. It is suggested that the subdued topography or inlet morphology may inhibit the development of secondary flows that have been recorded in previous flat-bed, rectangular open channel flows. A significant corollary of these results is that the morphological evolution of the pool–riffle sequence at high discharges may be controlled primarily by the downstream distribution of velocity and sediment transport with little role for lateral sorting and sediment routing by secondary flows. This paper also raises a number of issues that may be of use in future CFD modelling of three-dimensional flow in open channels within the geomorphological community.

Boundary Element Solution For The Cauchy Problem In Linear Elasticity

In this paper we investigate the solution of the Cauchy problem in linear elasticity using the iterative algorithm proposed by Kozlov et al. [1] for obtaining approximate solutions to ill-posed boundary value problems. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure.

The optimisation of reaction rate parameters for chemical kinetic modelling of combustion using genetic algorithms

A general inversion procedure for determining the optimum rate coefficients for chemical kinetic schemes based upon limited net species production data is presented. The objective of the optimisation process is to derive rate parameters such that the given net species production rates at various conditions are simultaneously achieved by searching the parameter space of the rate coefficients in the generalised Arrhenius form of the reaction rate mechanisms. Thus, the goal is to both match the given net species production rates and subsequently ensure the accurate prediction of net species production rates over a wide range of conditions. We have retrieved the reaction rate data using an inversion technique whose minimisation process is based on the Darwinian principle of survival of the fittest which has inspired a class of algorithms known as genetic algorithms. The excellent results presented here from our initial study are based upon the recovery of reaction rate coefficients for hydrogen/nitrogen/oxygen flames. The successful identification of the reaction rate parameters which correspond to product species measurement data from a sequence of such experiments clearly suggests that the progression onto other chemical kinetic schemes and the optimisation of higher-order hydrocarbon schemes can now be realised. The results of this study therefore demonstrate that the genetic algorithm inversion process promises the ability to assess combustion behaviour for fuels where the reaction rate coefficients are not known with any confidence and, subsequently, accurately predict emission characteristics, stable species concentrations and flame characterisation. Such predictive capabilities are of paramount importance in a wide variety of industries.

AN INVERSION METHOD WITH DECREASING REGULARIZATION FOR THE BACKWARD HEAT CONDUCTION PROBLEM

In this article an iterative algorithm is proposed for solving numerically the one-dimensional backward heat conduction problem (BHCP). The algorithm consists of using iteratively a least-squares fitting of the given data with a regularization term that decreases as the number of iterations increases. The algorithm is implemented using the boundary-element method (BEM). The convergence and the stability of the method are investigated for a severe test example, hence revealing the computational performance and limitations of the method proposed.

An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation

In this paper a new numerical method which does not require a regularization parameter is developed for solving the backward heat conduction problem (BHCP). The inverse and ill-posed BHCP for the heat equation is approximated with a convergent sequence of elliptic Cauchy problems for which the stable algorithm of the Kozlov et al. [1] type is accommodated. Further, a boundary element method (BEM) is developed for obtaining a stable solution of the BHCP.