2.9 Conservation and boundedness

The material derivative Eq. (2.14 ) includes the eqn term. However, the term can take for the form eqn, by applying conservation of mass Eq. (2.8 ), as shown in Sec. 2.8 . The two terms are related by the product rule

r (u ) = (r u) + u r : \relax \special {t4ht=

While the two terms eqn and eqn appear similar, they affect the solution of an equation is different ways.

PICT\relax \special {t4ht=

The eqn term maintains conservation in eqn. All variables are inside (to the right of) the divergence eqn, so when we integrate it over a volume eqn, it can be entirely transformed by Gauss’s theorem to an integral of the flux of eqn at eqn.

If we split eqn into two sub-volumes eqn and eqn, the surface integrals are also split over two surfaces eqn and eqn, where eqn is the part of the surface common to both volumes.

The fluxes at eqn for each sub-volume have equal magnitude but opposite sign, since their respective surface area vectors eqn point outwards from the volume in opposing directions. These fluxes cancel one another out, such that the sum of integrals over eqn and eqn is identically equal to the integral over eqn.

Thus, the term eqn ensures conservation in eqn across a surface separating regions of the flow domain. Conservation is guaranteed at all points in the limit eqn for any sub-volume.

The eqn form cannot transform to a surface integral so does not ensure conservation. Instead it ensures boundedness, as demonstrated by the solution of eqn in the following equation in one (eqn) dimension (1D)

@ @ ----+ ux----= 0: @t @x \relax \special {t4ht=
The solution to this equation has the form eqn. If eqn is uniform, then the profile of eqn does not change but simply translates at a speed eqn as shown in the left diagram. If eqn varies with eqn, the profile changes, e.g. flattens, as shown in the right diagram.

PICT\relax \special {t4ht=

In both cases, the eqn profile remains within the original bounds of eqn and eqn; the solution is said to be bounded. While the behaviour is illustrated in 1D, it extends to 3D.

Thus, the eqn term ensures boundedness. By contrast, the eqn term, while ensuring conservation, does not ensure boundedness when eqn. The two terms are connected by eqn which changes eqn due to expansion/contraction of the fluid, as discussed in Sec. 2.4 . Its effect is illustrated by flattening of the profile in the right diagram, since a non-uniform eqn corresponds to eqn.

Notes on CFD: General Principles - 2.9 Conservation and boundedness