# Statistical Uncertainty of DNS in Geometries without Homogeneous Directions
## 4 Conclusions  
We presented uncertainties of statistical quantities obtained in the DNS of channel flow, and of flow in a BFS. While the channel flow uncertainties are below a few percent, the uncertainties in the BFS at certain positions become higher. The highest relative uncertainties in the case of BFS are found near the stagnation zones, where velocities are low. As we have shown, spatial averaging can significantly reduce absolute statistical uncertainties. In case of channel flow, this is also carried over to relative statistical uncertainties.  
The uncertainty of individual components of average velocity, Reynolds stress tensor, average temperature, as well as the turbulent heat flux in case of BFS follows similar patterns as in the case of channel flow. The temperature fluctuations in the case of BFS show higher uncertainties in the recirculation region and close to the heat source.  
We provide a general estimator for statistical uncertainty of average velocity and temperature. Since it uses turbulence kinetic energy, temperature variance, average velocity and temperature, the estimator can be used to evaluate lower bound of the uncertainties of average velocity results in RANS simulations. Nevertheless, with an appropriate estimation of the decorrelation time of other variables, such as Reynolds stresses, the estimator could be used to predict relative uncertainties of that variable. For the DNS simulation, the estimator can be used to predict the averaging time needed for uncertainty to drop to required levels. Of course, this recommendation must be understood in a general way, since at some locations, the expected value of a velocity component might be zero and relative statistical uncertainty is meaningless at such points.  
We also observed that not every data sample from every time step is needed to produce accurate averages. The averages are accurate even if one includes only samples from every first or even tenth duration of the CFL time step. Obviously, the CFL time step is defined globally, for the whole domain. However, locally this period might even increase. Nevertheless, for accuracy of the uncertainty, we noticed that different periods of sampling could produce more discrepancies. Approximately by a factor of 2 or 3.  
**Author Contributions:** Conceptualization, I.T., C.F., J.O.; methodology, C.F., I.T., J.O.; simulation of channel flow, I.T.; BFS simulation, J.O.; writing--original draft preparation, J.O.; writing--review and editing, I.T., C.F.; visualization, J.O.; supervision, I.T.; project administration, I.T.; funding acquisition, I.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by Slovenian Research Agency, grants P2-0026 and L2-9210. Research is part of the SESAME project funded by the EURATOM Research and Training Programme (#654935) of the European Commission (H2020).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data from this article is available upon request to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.