Hybrid differencing scheme for multi-dimensional convection---diffusion

5.7.2 Hybrid differencing scheme for multi-dimensional convection---diffusion

The hybrid differencing scheme can easily be extended to two- and three- dimensional problems by repeated application of the derivation in each new co-ordinate direction. The discretised equation that covers all cases is given by

a P φ P =a W φ W +a E φ E +a S φ S +a N φ N +a B φ B +a T φ T (5.43)

5.8 THE POWER-LAW SCHEME

with central coefficient

a P =a W +a E +a S +a N +a B +a T + ∆F

and the coefficients of this equation for the hybrid differencing scheme are as follows:

One-dimensional flow

Two-dimensional flow

Three-dimensional flow

In the above expressions the values of F and D are calculated with the following formulae:

Modifications to these coefficients to cater for boundary conditions in two and three dimensions are available in the form of expressions such as (5.40).

5.8 The power-law scheme

The power-law differencing scheme of Patankar (1980) is a more accurate approximation to the one-dimensional exact solution and produces better results than the hybrid scheme. In this scheme diffusion is set to zero when

156 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS

cell Pe exceeds 10. If 0 < Pe < 10 the flux is evaluated by using a polynomial expression. For example, the net flux per unit area at the west control volume face is evaluated using

q w =F w [ φ W − β w ( φ P − φ W )] for 0 < Pe < 10 (5.44a)

where β

w = (1 − 0.1Pe w ) /Pe w

and

(5.44b) The coefficients of the one-dimensional discretised equation utilising the

q w =F w φ W for Pe > 10

power-law scheme for steady one-dimensional convection–diffusion

are given by

central coefficient: a P =a W +a E + (F e −F w )

and

D w max[0, (1 − 0.1| Pe w |) 5 ] + max[F w , 0] D e max[0, (1 − 0.1| Pe e |) 5 ] + max[−F e , 0] Properties of the power-law differencing scheme are similar to those of the

hybrid scheme. The power-law differencing scheme is more accurate for one-dimensional problems since it attempts to represent the exact solution more closely. The scheme has proved to be useful in practical flow calculations and can be used as an alternative to the hybrid scheme. In some commercial computer codes, e.g. FLUENT version 6.2, this scheme is available as a dis- cretisation option for the user to choose (FLUENT documentation, 2006).

5.9 Higher-order differencing

The accuracy of hybrid and upwind schemes is only first-order in terms of schemes for

Taylor series truncation error (TSTE). The use of upwind quantities ensures convection---diffusion

that the schemes are very stable and obey the transportiveness requirement, problems

but the first-order accuracy makes them prone to numerical diffusion errors. Such errors can be minimised by employing higher-order discretisation. Higher-order schemes involve more neighbour points and reduce the dis- cretisation errors by bringing in a wider influence. The central differencing scheme, which has second-order accuracy, proved to be unstable and does not possess the transportiveness property. Formulations that do not take into account the flow direction are unstable and, therefore, more accurate higher- order schemes, which preserve upwinding for stability and sensitivity to the flow direction, are needed. Below we discuss in some detail Leonard’s QUICK scheme, which is the oldest of these higher-order schemes.