Help for fourth order accurate convective schemes

Z.C.Wang

Hi everyone:I'm a Ph.D candidate working on direct numerical simulation of turbulence. These days I read Morinishi's paper "Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys. 143 (1998)" and got many useful information. The schemes proposed in his paper are very famous for the trait of fully conservation. I am interested in the fourth order accurate convective scheme in a staggered grid .But I have some problem with the equation 101. I try to derive the discrete formula of Eq.101 and write it in my code. However, I found the kinetic energy increase with time ! I have checked my code for several times and it works well when the convective term is discreted by second order scheme which is also proposed in his paper. I think I maybe derive the discrete form of Eq.101 in a wrong way. So could anyone please give me some advice of how to derive the discrete form of the convective term in a right way.