# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 3 Results  
### Uncertainties in Channel Flow  
Once the statistical steady state is recognized, the temporal and spatial averaging of the results was performed. A rather generally accepted rule of thumb says that the uncertainty of the basic statistical profiles in DNS should be less than \(\sim\)1%. However, a detailed study of [PERSON] [4] shows that it takes several hundred thousand time steps to achieve that target even in channel flow, where the statistical uncertainty is reduced also with the spatial averaging in two homogeneous directions. The corresponding number of time steps for the flows with a single homogeneous direction [17] is larger, and must be even larger in the geometries without homogeneous directions, where the only available type of averaging is averaging in time or ensemble averaging with several independent DNS runs performed in the same geometry.  
Profiles of the key turbulent statistics with quantified statistical uncertainty shown in Figure 3 compare two types of averaging: much lower uncertainty is achieved when temporal averaging is combined with the spatial averaging in two homogeneous directions (in addition, the symmetry of the channel in the \(y\) direction was also utilised with spatial averaging). However, as can be clearly observed, the reduction of statistical uncertainty with spatial averaging is not universal across the domain. As can be observed for the mean stream wise velocity component the benefit of spatial averaging is insignificant in the middle of the channel, while it reduces the uncertainty almost by an order of magnitude at around \(y^{+}\)\(\sim\)10. This could mean that our domain is not large enough. The effects of uneven reduction of statistical uncertainties across the domain are observed in all variables. The averaging time \(t^{+}\) was 54,000, and the time step was \(\Delta t^{+}\) = 0.027. This is about 2 million time steps; however, only every tenth time step was used for averaging.  
If the results shown in Figure 3 are applicable to an arbitrary turbulent flow without homogeneous directions, then reduction of statistical uncertainty requires typically millions of time steps and consequently large computational resources.  
For some variables, for example Root-Mean-Square (RMS) of the streamwise velocity component, the peaks of mean values correspond with peaks of statistical uncertainty. This results in constant relative statistical uncertainties. For mean streamwise velocity component, this percentage changes through the domain, but is around few tenths of percent. However, in the case of Reynolds stress \(\langle u^{\prime}v^{\prime}\rangle\), which is expected to be zero in the middle of the channel, the statistical uncertainty has a finite non-zero value, which causes the relative statistical uncertainty to diverge. If we discard such cases, we can claim that with temporal averaging only, the relative statistical uncertainties are also an order of magnitude higher. If we only observe the peaks of statistical uncertainties, however, the statistical uncertainty of shown results is around few percent without spatial averaging and around few tenths of percent with spatial averaging.  
The convergence rate of statistical uncertainties in channel flow DNS was explored by [PERSON] et al. [7] and more recently by [PERSON] and [PERSON] [18]. Additionally, [PERSON] and [PERSON] explained why statistical uncertainties of some global parameters converge with the rate, which is inversely proportional with averaging time.