Two-point, two-time closures applied to forced isotropic turbulence

The direct interaction approximation and the local energy transfer theory were used to calculate freely decaying turbulence for a range of initial Taylor–Reynolds numbers 3⩽Rλ(t=0)⩽129, and forced turbulence for evolved Taylor–Reynolds numbers Rλ(tevolved)≃88 and Rλ(tevolved)≃232. In the case of the freely decaying turbulence, the new feature of the work (as compared to McComb et al., J. Fluid Mech. 245 (1992) 279) was the ability to make comparisons with a direct numerical simulation at the same initial conditions. We were also able to obtain ‘experimental results’ from the simulation with an estimate of the experimental error. We present a small sample of results from an extensive investigation, sufficient to confirm that both closures agree with the simulation, to within the experimental error, in broad agreement with our earlier investigations. However, the main motivation of the work was to study the application of the closures to stationary turbulence and here we faced new problems such as how to ‘force’ the closures or the need to compute the time-history integrals of the closures over the long transient period during which the numerical simulation settles down. For this reason, our results should be seen as rather preliminary and tentative in nature. Essentially, we found that agreement between the closures and the simulation was not as good as for free decay for low-wavenumber behaviour but was quite good for high-wavenumber spectra and quantities determined by this region such as the Taylor microscale. We conclude that further work is needed to establish just what is a fair comparison between forced closure calculations and the analogous direct numerical simulation.