Notes on CFD: General Principles

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3.20 Other terms

In an equation for $\text{[math]}$ , there can be terms other than the time derivative, advection and Laplacian of $\text{[math]}$ , including:

a linear function $\text{[math]}$ where $\text{[math]}$ is a scalar coeﬃcient or ﬁeld;
a term without $\text{[math]}$ , sometimes including the derivative of another variable, e.g. $\text{[math]}$ .

The second example is simply discretised as an explicit gradient term as described in Sec. 3.15 . Derivative terms like this one are described in earlier sections and require no further discussion.

The ﬁrst example, the $\text{[math]}$ term, requires much more discussion, particularly relating to the possibility of implicit discretisation. Let us consider the following equation:

@ ---- c = 0: @t \relax \special {t4ht=

(3.33)

If $\text{[math]}$ is discretised implicitly, then the matrix would contain a zero diagonal coeﬃcient when $\text{[math]}$ , making it singular or non-invertible. If this occurs, $\text{[math]}$ is not present in the linear equation for the relevant cell, so it cannot be solved.

Therefore, such a term must be discretised explicitly when it has a negative sign (or positive on the right side of “=”), to ensure the matrix equation is solvable. The nature of Eq. (3.33 ) is that $\text{[math]}$ can only increase from an initial positive value.

Implicit discretisation of linear terms

Let us now consider the equivalent equation with a linear term that has a positive sign, i.e.

@ ----+ c = 0: @t \relax \special {t4ht=

(3.34)

With Eq. (3.34 ), $\text{[math]}$ can only decrease from an initial positive value but reaches a lower limit at $\text{[math]}$ . Like many scalar properties, e.g. $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , etc., $\text{[math]}$ may have a lower physical bound of 0, which the equation intentionally reﬂects.

It is important to maintain a physical bound of $\text{[math]}$ . Discretisation of Eq. (3.34 ) using the Euler time scheme Eq. (3.21 ) gives

= ---1--- - o; 1+ c t \relax \special {t4ht=

(3.35)

which ensures boundedness since $\text{[math]}$ . An equivalent explicit discretisation gives $\text{[math]}$ which is only bounded when $\text{[math]}$ , similar to the $\text{[math]}$ limit imposed by explicit discretisation of advection, described in Sec. 3.17 . To avoid this $\text{[math]}$ limit, we apply implicit discretisation to terms with a positive sign on the left side of “=”, where possible.

Even when the term is not linear in $\text{[math]}$ , it can be implemented as such by “dividing and multiplying by $\text{[math]}$ ”. For example, the $k$ turbulence model includes an $\text{[math]}$ term in the $\text{[math]}$ equation, i.e. $@k=@t +$ , ignoring other terms. Dividing and multiplying the $\text{[math]}$ term by $\text{[math]}$ gives