Abstract
In the previous paper regarding the numerical schemes for conservation of vorticity using Helmholtz element and for conservation of irrotational component, we have proposed a DNS for Leonard and Crosses terms in incompressible fluid. In the simulation, an implicit method has been used, dependent on the backward development scheme from time stage n+1 using Taylor series with terms of 4th order in time axis. All of time differentials in the Taylor series have been replaced with spatial differentials using Navier-Stokes equation, regarding in the appearance of 5th order multiple correlations. Thus this scheme is valid as a DNS for terms above. In this report, we tried to extend this scheme further with the help of Reynolds terms for the establishment of a DNS for Reynolds stresses. In the new scheme, we used an implicit method with terms of 2nd order in time axis for Reynolds stresses in the similar manner as described above. Accordingly, for the 2nd term, the transport equations of 3rd order multiple correlations of fluctuations and εij were required. Furthermore, in the 3rd order correlation of these transport equations, 4th order multiple correlations of fluctuations and others should be introduced. The main point in this report was that these values were calculated numerically by the direct use of the 4th order Taylor series as described.
