3.21Terms which change sign

An equation may include a function which can return a value which is positive or negative, e.g.

 (3.37)
Following Sec. 3.20 , we aim to avoid a singular matrix in the case and ensure the solution exceeds the lower bound of 0 when . The discretisation of is therefore treated implicitly or explicitly within each cell based on
 (3.38)

Maintaining lower and upper bounds

The solution variable could have a physical bound at any low value and/or high value . We can treat the function in Eq. (3.37 ) as follows to maintain boundedness:

 (3.39)
Here, and are calculated using the current values of .

If we examine the situation with , Eq. (3.37 ) would be discretised as

 (3.40)
When we discretise implicitly alongside the Euler time scheme, Eq. (3.40 ) becomes:
 (3.41)
The solution for decreases when , but cannot fall below the bound within a solution step. In the limit that , the solution for no longer decreases.

An equivalent analysis shows Eq. (3.40 ) maintains boundedness for the upper bound . In that case, note that is negative so the term in can still be treated implicitly.

Fields bounded by 0 and 1

Many properties are expressed as a fraction, e.g. concentration of a chemical species, so are bounded by 0 and 1. When that happens, Eq. (3.39 ) simpliﬁes to

 (3.42)
Here, and are calculated using the current values of .

It is particularly important to maintain boundedness of a ﬁeld with 0-1 bounds when it is used to calculate another property which has a physical bound. For example, in a system of ﬂuid phases, ﬂuid can be calculated by from phase fractions and densities .

A small amount of unboundedness in , in the phase with highest , causes large unboundedness in , e.g. in 2 phases with and , the calculated with .

Notes on CFD: General Principles - 3.21 Terms which change sign