3.22 Bounded advection discretisation

Sec. 2.9 described the conservation and boundedness qualities of the advective derivatives eqn and eqn, respectively. According to Eq. (2.31 ), the terms are equivalent when eqn, i.e. when mass conservation Eq. (2.8 ) is satisfied in the case when eqn 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 eqn.

Discretisation of an advection term of the form eqn as described in Sec. 3.9 can cause unboundedness in eqn. If eqn has a physical bound that is violated, the solution rapidly becomes unstable.

The addition of the eqn term ensures boundedness while eqn, at the expense of conservation. Once eqn, the solution is both bounded and conservative.

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

This additional term is implicit in eqn, so contributes eqn to the diagonal coefficients of the matrix. If eqn 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 eqn. If that term uses the upwind scheme, the discretisation can be split into a contribution from positive outgoing fluxes eqn and negative incoming fluxes eqn. In extensive form (i.e. scaled by eqn, see Sec. 3.24 ), using values in the cell of interest eqn and neighbouring cells eqn, the terms are
X + + X X + + X : f f N f f |--------------------------{z--------------------------} |------------------------{z------------------------} r (u ) (r u) \relax \special {t4ht=
The eqn terms cancel, leaving negative coefficients for neighbouring cells (since eqn 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 eqn which represents the fraction of the unburnt fuel mixture.

PICT\relax \special {t4ht=

The equation for eqn includes the source term eqn, where eqn is a calculated flame speed. Its inclusion could cause the solution of eqn to fall below its lower bound of 0.

Multiplying and dividing by eqn changes the term to eqn, where eqn and unit vector eqn. In this form, the term represents the non-conservative advection of eqn by a flame velocity eqn moving in the direction eqn, 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