2.9Conservation and boundedness

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

 (2.31)

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

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

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

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

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

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

 (2.32)
The solution to this equation has the form . If is uniform, then the proﬁle of does not change but simply translates at a speed as shown in the left diagram. If varies with , the proﬁle changes, e.g. ﬂattens, as shown in the right diagram.

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

Thus, the term ensures boundedness. By contrast, the term, while ensuring conservation, does not ensure boundedness when . The two terms are connected by which changes 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 corresponds to .

Notes on CFD: General Principles - 2.9 Conservation and boundedness