Notes on Computational Fluid Dynamics: General Principles

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3.22 Bounded advection discretisation

Sec. 2.9 described the conservation and boundedness qualities of the advective derivatives $\text{[math]}$ and $\text{[math]}$ , respectively. According to Eq. (2.31 ), the terms are equivalent when $\text{[math]}$ , i.e. when mass conservation Eq. (2.8 ) is satisﬁed in the case when $\text{[math]}$ constant.

During computation of an incompressible ﬂow, numerical error can be signiﬁcant, in particularly at the beginning of a steady-state calculation. The governing equations are not satisﬁed, including mass conservation, so $\text{[math]}$ .

Discretisation of an advection term of the form $\text{[math]}$ as described in Sec. 3.9 can cause unboundedness in $\text{[math]}$ . If $\text{[math]}$ has a physical bound that is violated, the solution rapidly becomes unstable.

The addition of the $\text{[math]}$ term ensures boundedness while $\text{[math]}$ , at the expense of conservation. Once $\text{[math]}$ , the solution is both bounded and conservative.

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

(3.43)

This additional term is implicit in $\text{[math]}$ , so contributes $\text{[math]}$ to the diagonal coeﬃcients of the matrix. If $\text{[math]}$ is positive, we might expect a singular matrix, as discussed in Sec. 3.20 . However, the decrease to the diagonal coeﬃcient is matched by an increase from the discretisation of $\text{[math]}$ . If that term uses the upwind scheme, the discretisation can be split into a contribution from positive outgoing ﬂuxes $\text{[math]}$ and negative incoming ﬂuxes $\text{[math]}$ . In extensive form (i.e. scaled by $\text{[math]}$ , see Sec. 3.24 ), using values in the cell of interest $\text{[math]}$ and neighbouring cells $\text{[math]}$ , the terms are

X + + X X + + X : f f N f f |--------------------------{z--------------------------} |------------------------{z------------------------} r (u ) (r u) \relax \special {t4ht=

The $\text{[math]}$ terms cancel, leaving negative coeﬃcients for neighbouring cells (since $\text{[math]}$ is negative). The diagonal coeﬃcient is then equal to the sum of magnitude of those neighbour cell coeﬃcients, resulting in an invertible matrix.

Capturing physics

Boundedness and conservation can be compromised when a term in an equation does not capture the physics of the problem. An example from ﬂame speed combustion modelling uses a parameter $\text{[math]}$ which represents the fraction of the unburnt fuel mixture.

$PICT\relax \special {t4ht=$

The equation for $\text{[math]}$ includes the source term $\text{[math]}$ , where $\text{[math]}$ is a calculated ﬂame speed. Its inclusion could cause the solution of $\text{[math]}$ to fall below its lower bound of 0.

Multiplying and dividing by $\text{[math]}$ changes the term to $\text{[math]}$ , where $\text{[math]}$ and unit vector $\text{[math]}$ . In this form, the term represents the non-conservative advection of $\text{[math]}$ by a ﬂame velocity $\text{[math]}$ moving in the direction $\text{[math]}$ , which captures the physical nature of combustion. Boundedness can then be maintained by a suitable choice of advection scheme.

Notes on CFD: General Principles - 3.22 Bounded advection discretisation