General comments on the QUICK differencing scheme The QUICK differencing scheme has greater formal accuracy than the

5.9.4 General comments on the QUICK differencing scheme The QUICK differencing scheme has greater formal accuracy than the

central differencing or hybrid schemes, and it retains the upwind-weighted characteristics. The resultant false diffusion is small, and solutions achieved with coarse grids are often considerably more accurate than those of the upwind or hybrid schemes. Figure 5.20 shows a comparison between upwind and QUICK for the two-dimensional test case considered in section

5.6.1. It can be seen that the QUICK scheme matches the exact solution much more accurately than the upwind scheme on a 50 × 50 grid.

Figure 5.20 Comparison of QUICK and upwind solutions for the 2D test case considered in section 5.6.1

The QUICK scheme can, however, give (minor) undershoots and over- shoots, as is evident in Figure 5.20. In complex flow calculations, the use of QUICK can lead to subtle problems caused by such unbounded results: for example, they could give rise to negative turbulence kinetic energy (k) in k– ε model (see Chapter 3) computations. The possibility of undershoots and overshoots needs to be considered when interpreting solutions.

5.10 TVD schemes Schemes of third-order and above have been developed for the discretisation

of convective terms with varying degrees of success. Implementation of boundary conditions can be problematic with such higher-order schemes. The fact that the QUICK scheme and other higher-order schemes can give undershoots and overshoots has led to the development of second-order

5.10 TVD SCHEMES 165

schemes that avoid these problems. The class of TVD (total variation dimin- ishing) schemes has been specially formulated to achieve oscillation-free solutions and has proved to be useful in CFD calculations. TVD is a property used in the discretisation of equations governing time-dependent gas dynamics problems. More recently, schemes with this property have also become popular in general-purpose CFD solvers. Fundamentals of the development of TVD methodology involves a fair amount of mathematical background. However, the ideas behind TVD schemes can be easily illus- trated in the context of the discretisation practices presented in the previous sections by considering the basic properties of standard schemes and their deficiencies.

As discussed earlier, the basic upwind differencing scheme is the most stable and unconditionally bounded scheme, but it introduces a high level of false diffusion due to its low order of accuracy (first-order). Higher-order schemes such as central differencing and QUICK can give spurious oscilla- tions or ‘wiggles’ when the Peclet number is high. When such higher-order schemes are used to solve for turbulent quantities, e.g. turbulence energy and rate of dissipation, wiggles can give physically unrealistic negative values and instability. TVD schemes are designed to address this undesirable oscil- latory behaviour of higher-order schemes. In TVD schemes the tendency towards oscillation is counteracted by adding an artificial diffusion fragment or by adding a weighting towards upstream contribution. In the literature early schemes based on these ideas were called flux corrected transport (FCT) schemes: see Boris and Book (1973, 1976). Further work by Van Leer (1974, 1977a,b, 1979), Harten (1983, 1984), Sweby (1984), Roe (1985), Osher and Chakravarthy (1984) and many others has contributed to the develop- ment of present-day TVD schemes. In the next section we explain the funda- mentals of the TVD methodology.