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From D´Alascio et al. (2004). A middle plane section of an helicopter fuselage with structured and unstructured grids. 3). As the mesh point values are the sole quantities available to the computer, all mathematical operators, such as partial derivatives of the various quantities, will have to be transformed, by the discretization process, into arithmetic operations on the mesh point values. This forms the content of Part II, where the different methods available to perform this conversion from derivatives to arithmetic operations on the mesh point values will be introduced.
For any quantity U , physical assumptions must provide definitions for the fluxes and the source terms, in function of other computed variables. 1, we have not provided any specific information concerning the fluxes, except for the fact that they do exist for any conserved quantity U . However, we can now be more specific and look closer to the physics of transport of a quantity U in a fluid flow. The fluxes are generated from two contributions: a contribution due to the convective transport of the fluid and a contribution due to the molecular agitation, which can be present even when the fluid is at rest.
Differential form of a conservation law An alternative, local differential form of the conservation law can be derived by applying Gauss’ theorem to the surface integral term of the fluxes and the surface sources, assuming that these fluxes and surface sources are continuous. Gauss’ theorem states that the surface integral of the flux is equal to the volume integral of the divergence of this flux: F · dS = ∇ ·Fd S for any volume , enclosed by the surface S, where the gradient or divergence operator ∇ is introduced.