The finite volume method for convection diffusion problems

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The finite volume method for convection -diffusion problems INTRODUCTION In problems where fluid flow plays a significant role we must account for the effects of convection. Diffusion always occurs alongside convection in nature. The steady convection–diffusion equation can be derived from the transport equation for a general property  by deleting the transient term div(u) = div(  grad ) + S Formal integration over a control volume gives ∫n . (u)dA = ∫n . (  grad )dA + ∫ S dV A A CV The left hand side gives the net convective flux and the right hand side contains the net diffusive flux and the generation or destruction of the property  within the control volume. The principal problem in the discretisation of the convective terms is the calculation of the value of transported property  at control volume faces and its convective flux across these boundaries. For the diffusion problem with source term, the central differencing method of obtaining discretised equations is suitable. It would seem obvious to try out this practice, which worked so well for diffusion problems, on the convective terms. However, the diffusion process affects the distribution of a transported quantity along its gradients in all directions, whereas convection spreads influence only in the flow direction This crucial difference manifests itself in a stringent upper limit to the grid size, which is dependent on the relative strength of convection and diffusion, for stable convection–diffusion calculations with central differencing. . In the current analysis no reference will be made to the evaluation of face velocities. It is assumed that they are ‘somehow’ know Steady one- dimensional convection and diffusion In the absence of sources, steady convection and diffusion of a property  in…