3.9 Advection discretisation

In Sec. 2.8 , we described advection terms as those of the form eqn or eqn. The inclusion of velocity eqn within the divergence gives the term particular characteristics that require special treatment in discretisation.

PICT\relax \special {t4ht=

Following the finite volume principles outlined in Sec. 3.1, the discretisation approximates the surface integral by a summation over faces by

pict\relax \special {t4ht=

The discretisation requires calculation of the volumetric flux eqn (see Sec. 2.3 ). In the case of eqn included in the advection term eqn, eqn is the mass flux eqn.

In the flux calculation, the interpolation of eqn at cell centres to eqn at faces uses the linear scheme of Sec. 3.7 . Similarly, the linear scheme is used for the interpolation eqn.

The critical issue — one of the most important in CFD numerics — is how to express our advected property eqn at a face in terms of values eqn in neighbouring cells.

Advection scheme introduction

The advected property eqn is transported in the direction of flow velocity eqn. Interpolation to the face of eqn usually involves the flow direction. In the graphic below a face f is positioned between two cells. Based on the flow direction, the cells are labelled upwind ‘U’ and downwind ‘D’.

PICT\relax \special {t4ht=

The linear interpolation scheme, 5 described in Sec. 3.7 , does not use the flow direction but expresses eqn in terms of adjacent cells. At first sight, this choice of scheme is logical when considering accuracy. However, for advection, the linear scheme tends to generate unbounded solutions which are unstable.

The upwind scheme simply represents eqn by the value in the upwind cell eqn. It makes sense from a physical perspective since particles of fluid in the upwind cell are destined to travel to the face, transporting property eqn with them.6

While the linear scheme is generally unbounded, the upwind scheme exhibits poor accuracy. In following section we explore the behaviour of upwind before looking at schemes that offer greater accuracy while attempting to maintain boundedness.

5also known as the central difference scheme, particularly in the context of advection.
6or, to quote Brian Spalding: “the wind from a pigsty always stinks”.

Notes on CFD: General Principles - 3.9 Advection discretisation